This is my working hypothesis - the way I see it as of this moment!! 

   1659 John Moore  baptized at St Bride's in London,  England 
   ca 1680 - John Moore went from England to South Carolina 
 ?  ca 1680 - John Moore married Katherine
 ? 1681 - daughter Frances is born.
    1683 - and was Secretary of South Carolina. 

  ca 1685 John Moore married  Lady Rebecca Axtell in Charlestown, SC, daughter of Landgrave Daniel Axtell, who was son of the Gov of Kilkenny, Lt Col Daniel Axtell.
   ca 1697 he moved to Philadelphia and became "practitioner of law."
    1700 -  John was Attorney-General for the King [19 10 mo 1700]
      Deputy-Judge under Quarry
   1701 -    Register-General of Wills May 17, 1701 to 27 June 1704.
   1704 to 1732 -   Collector of Customs July 24, 1704 to 1732.
       he lived on the west side of Second Street in the first house north of the parsonage of Christ Church, which parsonage adjoined the Church on the north, his garden, etc. being at the corner of Garden [now M'Comb Alley] 
   25 Nov 1732 -   John Moore died  in Philadelphia.
    7 Dec 1732.   He was buried in the middle aisle of Christ Church, Philadelphia 
 (of which he was one of the founders and a vestryman), opposite his pew in the main
aisle in front of the pulpit.)
      His wife's name was Rebecca Axtell.  Rebecca Moore's will was dated Nov. 23, 1749, proved Jan 15, 1750, (son William sole executor).   She was buried 'in St. Peter's Church in the Great Valley,' Chester Co, Penn.

  Children of John Moore and Rebecca:
 1. Col. John Moore 11 Aug 1686 SC - 29 Oct 1749 NYC age 64
         Alderman, member of Legislature, Col of Regiment, Member of Provincial Council NY
         will dated 4 Sept 1748, buried in his vault in Trinity Churchyard
       married Frances Lambert Apr 1692 - 17 March 1782 
2. Thomas Moore 1689 SC - 1769 Little Britain, London, Eng.
          Rector of Little Britain in London 1753. Rector of Christ Church in Kent 1755
3. Rebecca Moore ca 1692 -
      married 28 Aug 1709 Phil. PA  John Evans 
            Ex-governor of Pennsylvania, held the office about four years, after surrendering his office to Gookin remained for a time in Philadelphia but at length retired to Denbigh, Wales.
 In Nov 1716 purchased of his former companion young William Penn 2000 acres of the manor of Steryning on Brandywine Creek and the following March gave John Moore his father in law power to sell the same. But in 1731 as John Evans of Pentry Manor, County Dunbigh, he declared before a master of chaucery that he never authorized Moore to sell any land.
4. Mary Moore  ca 1694 - Nov 1735 in Phil.
       married ca 1715 Peter Evans d. 1745 in Phil.
            Register General in Philadelphia [a cousin of John Evans and Sheriff of Philadelphia.]
5. William Moore 6 May 1699 Phil - 30 May 1783 Chester Co, PA
            home known as "Moore's Hall"  -- Chief Justice of Chester County
          buried at St. David's, Delaware Co, PA; will leaving all to wife dated 24 May 1783
       married 1722 Wilhemina Weems [Wemys] ca 1704 Scotland - 6 Dec 1784
           daughter of James, the 4th Earl of Weems, descended from MacDuff, Earl of Fife
6. Daniel Moore 
       married in Barbados a lady of great fortune
          he went to Barbadoes thence to England residing at Great Marlow, MP for Great Marlow
           many years member of Parliament for Great Marlow
           his daughter Frances being the wife of Chancellor Erskine
7. Richard Moore ca 1709 - 1738 bur: 29 July 1738 Kingston, Jamaica  in Barbadoes
     married at St. Andrew 14 Apr 1737 Rachel Martin bapt St. Andrew 1707 -
         dau of John Martin died and buried 27 Dec 1710 and wife Frances Wastel died and buried 16 Dec 1714
8. Somerset Moore bapt 19 July 1711 at 5 days of age - 1 Oct 1712
9. Charles Moore  buried 17 Aug 1712

    Frances Wastel ca 1681 - 1714 age 33       St. Andrew Parish Records 1664-1807, Jamaica
      married 27 Feb 1700/01 John Martin ca 1678 - bur 27 Dec 1710 Kingston, Jamaica age 32
     1. Wastel Martin son bapt St. Andrew 23 Jan 1702/03
     2. Mary Martin bapt SA 18 Mar 1703 ?
     3. John Martin bapt SA 8 June 1706 -
     4. Rachel Martin bapt SA 1707 -   no record of children or remarriage
          married 14 April 1737 Richard Moore ca 1709 - 1738
     5. Griswell dau bapt 8 June 1710 -
   Frances married 2nd 30 Oct 1712 John Terry
 -- in 1738 -- Richard Moore ca 1709 - 1738 a son of John Moore and his wife Rebecca Axtell died in Jamaica. In St. Andrew's Parish, Kingston, Jamaica, there is a monumental inscription engraved with the names of Richard Moore, Frances Martin and her husband John Martin, and a coat of arms showing the ten crosses, crosslet, impaled with the arms of Martin [3 bendlets, a chief erm]. John Martin died in 1710 age 32 and Frances died 1714 age 33. [Who were John and Frances Martin? . . .]

"'Hon.' John Moore, Esq., Barrister-at-Law, 1658 Eng.-1732 Philadelphia  of London, of 'Moore's Hall in Charleston SC', of Moore Hall, co. Chester, Penn., and of Philadelphia, Penn., known to some as Sir John Moore, and called in Pennsylvania, Hon. John Moore, on account of being designated as 'Hon. John Moore, Gent..' in his several conveyances of realty, recorded in Christ Church deeds in 1695 and 1702;  Colonist in 1680, Secretary of the Province of Carolina, south and west of Cape Fear, June 21, 1682;  Receiver General and Escheator Gen. of same, Sept. 29, 1683; Deputy to Lord Proprietor and Count Palatine, Sir Peter Colleton, 2nd Baronet, (son of Sir John Colleton, 1st Baronet, of His Majesty's Foot, one of the original grantees under the charter from King Charles II upon his restoration), June 3, 1684, thus constituting him a member of the Governor's Grand Council, (Governor Joseph Moreton's first term),  m. in 1685 Lady Rebecca ( -1745), daughter of the Landgrave Daniel Axtell, hereditary Peer of the Dominion of Carolina under under the Government of Lords, and Lady Rebecca (Holland), his wife, and granddaughter of Lieutenant-Colonel Daniel Axtell of the Parliamentary Army, Governor of Kilkenny in Ireland, and Commander of the Guard of the High Court of Justice in Westminster Hall during the trial of King Charles I.
 Between 1687 and 1690, he moved from Charleston, S.C., to Philadelphia, Penn., (with Col. Robert Quarry, Judge of the Court of Vice Admiralty, (1695-1703)), and held office as follows: Crown Advocate and Deputy Judge of the Court of Vice Admiralty, prov. of Penn., in 1695; Attorney General for the prov. and territories of Penn., (May 19, 1698-1700);  Registrar Gen. of Penn., ;(Jan. 1, 1693- March 29, 1703);  Registrar of Wills, (May 17, 1701-June 27, 1704); also Collector of His Majesty's Customs for the Port of Philadelphia, June 24, 1703 or 04, which office he held under three successive princes, (photographic copy of his third original commission as Collector as above, dated first year of King George II, Dec. 19, 1727, and signed by William Manley, T. Walker, Walter Yongue and John Hill, for his Majesty's Customs Commissioners, is in possession of his great-great-great-great-grandson,. Capt. H. H. Bellas, U.S.A., Phila),  which last office he held until Nov. 25, 1732, when he died and was buried in Christ Church, Phila., (of which he was one of the founders and a vestryman), opposite his pew in the main aisle in front of the pulpit. Dec. 7, 1732."
References:
 APPLETON'S ENCY. OF AMER. BIOG., ed 1888. Col. So. Car.Hist. Soc., vol 1, pages 109-10. 
Ibid page 111,
 Ibid vol and page, 
also mss. copy of Council Journal, pages 60, 69, 
Ramsey's So. Car., 
Old Mixon's Carolina, 
Carroll's Hist. Coll. vol. 2, pages 410. 
The references to 'Coll. So. Car. Hist. Socy.' are from the publication of lists of documents relating to South Carolina in the State Paper Office, London, Eng.., also upon authority and personal researches of Rev. Robert WILSON, D.D., President of the Huguenot Society ot South Carolina, of Charleston, S.C. 
Lady Ann BOONE's will (daughter of Landgrave Daniel AXTELL), So. Car. Probate Office, W. B. of 1751, Hume's England,
Pepy's Diary 1660,
 Penn State Archives, vol 9, 2nd series edition 1880, page 627.
 Penn. Gazette, Nov. 30, 1732, 
and Christ Church Burial Records; 
also Scharf and Westcott's Hist. of Phila., vol III, pae 1804. 
Logan's, vol I, page 309. Hon. John MOORE's will, dated Nov. 16, 1731, proved Dec. 13, 1732, (wife sole executrix). 
Lady Rebecca MOORE's will dated Nov. 23, 1749, proved Jan 15, 1750, (son William sole executor). 
Horace Wemyss SMITH's Life of Rev. William SMITH, D. D., appendix, vol. II, page 494,
 Foster supra, John MOORE, genealogist. 
Life of AXTELL, with portrait, London, 1660, Lee supra.
Note.--Rebecca is variously written in original wills, as 'Rebekah' and 'Rebeccah.' 
The descendants of the above four children of Hon. John MOORE are traced separately. 
Lieut.-Col. Daniel AXTELL, Governor of Kilkenny under the Protectorate, an active participant in Pride's Purge and preceding events leading to the execution of Charles I, was excepted from the Bill of Amnesty in July 1660, and the pardons granted the following August. He was arraigned at the Old Bailey, charged with commanding the soldiers at the High Court of Justice, threatening to shoot Lady FAIRFAX for interrupting the proceedings, and personal insults to the King.
 To the first, he plead the order of the Commander-in-Chief, 
  to the second, the exigencies of the occasion,
  the third, he utterly denied. 
The sentence for treason was fully carried out and his head was set on the further end of Westminster Hall. (Ludloe's Memoirs, State Trials).
  The Landgrave AXTELL, son of the regicide (will proved in London circa 1692), had several children, among them - Holland, hereditary Peer, d. unm. Anne. Elizabeth m. to the Landgrave BLAKE, Rebecca and Sybil."

Grandchildren of John Moore and Rebecca:

Children of John Moore and Frances Lambert:
1. Frances Moore  1715 - 1805
       married Samuel Bayard ca 1715 Throgg's Neck, Westchester, New York
2. Rebecca Moore 1717 - sp in 1748    dsp
3. John Moore  1719 - bur 9 Aug 1747  dsp
4. Susanna Moore 1720 - dy  
5. Thomas Moore  1721 - dy
6. Peter Moore 1721 - dy  twin of Thomas
7. Thomas Moore 1722 - 1784 New London CT
         lived at White Hall, NYC  until it was burned by the British during the Rev.
      married 1744 Elizabeth Channing 1728 - 1805
8. Peter Moore 1722 - dy twin of 2nd Thomas #7
9. Richard Moore 1724 -  dsp 
10. Susannah Moore 1725/6 -  sp in 1748
11. Daniel Moore 1727 - dy  twin with Lambert 
12. Lambert Moore 1727 - 1805 
        was Deputy Sec of New York, Controller of Customs, Clerk of Trinity Church,
           interred in family vault at Trinity
      married 31 Jan 1757 Jane Holland
             dau of Edward Holland and Magdalene Bayeux;  g dau of Henry [Holland]
13. Daniel Moore 1728 - dy
14. Daniel Moore 1729 -1789  dsp
15. William Moore 1730 -  dsp
16. Charles Moore ca 1732 -
17. Stephen Moore 19/30 Oct 1734 NYC - 20 May 1799 Mt Tirzah, Person Co, NC
       married in Quebec 25 Dec 1768 Griselda Phillips 18 Feb 1748/9 Boston - 
18. Ann Moore  1738 - ca 1826 
"in early life she passed seven years in England under the patronage of her uncle,
Daniel Moore, Esq., M.P. 

Children of Thomas Moore and ------ :
Rev. Thomas Moore I, D.D., (1689 Penn.-1769 Eng.), Fellow of Trinity College, Cambridge, and graduate of the University of Oxford, (B.A. 1712, M. A. 1716, D. D. 1733), was Chaplain to the Dean and Chapter of Rochester Cathedral, during the incumbency of the Rt. Rev. Francis Atterbury, D. D., Lord Bishop of Rochester, who was banished for his attachment to the House of Stuart. Dr. Moore was taken into custody with Bishop Atterbury, but his innocence being quickly proven, he was promptly released. He was subsequently Rector of Chiselhurst, co. Kent, and Little Britain, London. Some years before his death, he declined an offer to promotion to the Bishop's Bench, feeling and saying in reply to the proposal that he had been too long and unjustly neglected.
  He is said to have had nine children, among them. 
1. Rev. Thomas Moore II 1738 - 1823 aged 85
  "who matriculated at Worcester College, Oxford. May 26, 1753, age 15, and took his B. A. in 1757 and M.A. in 1759, and was rector of North Cray, co. Kent in 1765, and Foot Cray in 1768 till 1823, when he died, aged 85."  --  "held the parish of North Craig in Kent, England more than fifty years, his portrait is in possession of the widow of Thomas William Moore"
2. Rev. Charles Moore  1740 -
 ".. rector of Caxton and Vicar of St. Nicholas, Rochester, co. Kent. Two volumes of his sermons were published circa 1820 by one of his sons. "
   "held two livings in Kent for many years had wife and nine children [letter in possession of T W Moore dated Aug 25 1755] had grandson Capt. Charles Moore of British Army in a letter from Capt. Charles to the father of T W Moore is appears there was only one of his children living, a lady of 80 who had three children living date of letter, London, June 5 1826."
      a. Major Charles Moore 1760 - 1856 aged 96
        "who died at his residence in the castle yard, Windsor, in the spring of 1856, aged 96. The press of the day stated that 'he had been Governor of the Military Knights of Windsor for 22 years with the same high honor and ability by which he was characterized in his long military service. He died universally esteemed and regretted.' T. W. C. Moore of New York called upon him at Windsor Castle on his last trip to England."
3. John Moore of Vale Place, Hammersmith
      " druggist, retired from business, and now (1824) living at Vale Place.
4. William Moore of London
         "jeweler, residing in London (1824)
5. 6. 7. 8. 9.

Children of Rebecca Moore and John Evans:
 Rebecca m. in 1709 to Colonel John Evans, Lieutenant Governor of Penn. Issue. Two daughters.. This family migrated to Denbigh, Wales. Col. and Mrs. Evans were in London at the probate of the will of Thomas Moore, Esq., registrar of the diocese of Rochester, Jan. 16, 1734.
1. daughter Evans ca 1710 -
2. daughter Evans ca 1712 -

Children of Mary Moore and Peter Evans:
1. Rebecca Evans
      married Peter Robertson
2. John Evans
3. Margaret Evans
4. Mary Evans 

Children of William Moore and Wilhemina Weems of Phila.:
1. Rebecca Moore 21 Feb 1724/25 - dy
2. William Moore 5 Oct 1726 -
      [many years a partner of his uncle Richard in Barbados]
     a. William Sturge Moore
           married Williamina daughter of his uncle Thomas William
3. Wiliamina Moore 21 Feb 1726/27 -1809
      married 1748 Dr. Phineas Bond 18 June 1717 - 2 June 1773
       a. Wilhemina Bond 26 Feb 1753 - 9 Sept 1837
            married Thomas Cadwalader 10 Jan 1741/42 - 10 Feb 1786
            i. daughter married Mr. Erskine, son of the Lord Chancellor of England
4. John Moore 1 Oct 1729 - dy
5. John Moore 21 Jan 1730/31 -
6. Rebecca Moore 21 Feb 1732/33 - 6 Dec 1784
      married 3 July 1758 Rev Dr William Smith
           Provost of University of Pennsylvania
7. Thomas William Moore 12 June 1735 -
       auctioneer at New York in full business and highly esteemed
      married Ann Langdon [widow of Dr. Ayscough]
      a. Thomas William Moore
                  British Agent in New York
      b.  Williamina Moore
             married Sturge Moore
      c. Janet Moore
            married Capt Jacob Jones
8. Margaret Moore 26 Mar 1738 -
9. Mary Moore 8 July 1741 - 8 July 1741
10. Anne Moore 4 Oct 1742 - 20 Dec 1810 Dover, Del.
     married 2 June 1774 Charles Greenberry Ridgely  26 Jun 1737 - 25 Nov 1785
        "MD"
11. Frances Moore 10 Mar 1744/45 -
12. James Weems Moore 22 July 1747 -
13.  unnamed Moore 11 Dec 1749 - 11 Dec 1749

 "NO evidence has been found of another daughter marrying Thomas Bond."

  Child of Daniel Moore and lady of Barbadoes:
1. Frances Moore d. 22 Dec 1805 Hampstead, Middlesex, Eng.
     married 29 March 1770 Thomas Erskine 10 Jan 1749/50 - 17 Nov 1823
      [he had son who married Miss Cadwallader (see above)]
2. John Moore

Ref:
1. p 475/6 in Rev. John Moore of Newtown, LI by James W Moore [1903]
2.  SIX CENTURIES OF MOOR DE FALLEY by David Moore Hall, printed
1904 by O.E. FLANHART Printing Co., Richmond, VA.
3. Memoirs of John Moore, Esq. and The Case for the Corrected Lineage of Hon. John Moore
[Originally published in Moore/Stanford/Webb Chronicles Vol 4, No 1, 1996 edited by Terry Bradshaw O'Neill]