This is my working hypothesis - the way I see it as of this moment!! 
Dr. G C Moore we are told claimed kin or some kind of a connection to this family but I have been unable to find the connection. --notes on family

    Stephen Moore was married 25 Dec. 1768 in Quebec, Canada, by Rev. Mr. Montgomery, Chaplin of the King's 10th Regiment, to Grizey Phillips [of Boston, Mass].

   Stephen Moore was born 30 Oct 1734 in New York City.  He was the 17th of 18 children of John Moore [1686 SC - 1749 New York City] and Frances Lambert [1692 - 1782]. His father was  "Col. John Moore, a leading merchant in the early days of New York city"

   Grizey Phillips was born 18 Feb 1749 in Boston, MA daughter of Capt John Phillips (1710-1773) and his wife. She died 14 Jan 1822 in Person County, NC.

  Stephen was "brought up in business by John Watts." Typical of the times he would have begun his training ca age 14. John Watts was a prominent merchant and a provision contractor to the British army during the French and Indian War. 

  In 1757 Stephen Moore was commissioned a Lieutenant in the New York Provincial troops and participated in the French and Indian War. 

 -- An enduring family legend --
   John Moore, Esq [1745-1828] in one of his memoirs [JAH 4-#1, 1910] gives an account of his uncle Stephen Moore's feat  in mid-winter about the year 1762/3  of delivering dispatches from General Haldimand in Quebec to Sir Jeffrey Amherst in New York City, in TEN day's time. With two Mohawk guides he went overland on snowshoes, then canoed down Lake Champlain & George, ice-skated down the Hudson as far as Yonkers, then on foot the last 20 miles to New York City.  All the principals can be placed in the right places at the right time. It is known that Stephen often traveled this route in later years and the timing would have afforded him the opportunity of visiting his family after having been away as a participant of several years in the French and Indian War.  from article by Terri Bradshaw O'Neill 

   At the conclusion of hostilities, Stephen was in Quebec, Canada, where he settled for a period of about ten years and was engaged in trade, shipping and mercantile business. Leaving Quebec in late 1770, he settled his family at West Point, NY at the estate he inherited at the death of his father. Now-- the move to NC may have been a result of the "rumor of war" and the strategic location of his home. By Sept 1776, his family was already established in Orange County and by 1777 he was acquiring land and building his estate, Mt. Tirzah in Caswell [later Person] county.  from info provided by Terri Bradshaw O'Neill

  In 1779 he was given charge of a Regiment of NC State Troops. He was in the 1st Battle of Camden [SC].
 " Moore, Stephen [NC]  Lieutenant-Col. NC Militia -- taken prisoner at Charleston, 12th May 1780."

 "The elder Stephen Moore was the owner of West Point and sold it to the US Government in 1790."Page 34 - 

   Stephen Moore died at Mt Tirzah, Person County, NC  29 Dec 1799.
       Moore Family Bible #1900-Webb Papers, Southern Historical Col, UNC-Chapel Hill

 "Gen. Stephen Moore, of White Hall, Moore's-Folly-on-the-Hudson, N. Y., and Mount Tirzah, N. C. (1734-1799), Lieut. DeLancey's Brigate, British Army, (war 1756); afterwards Paymaster to the army in Canada,   During the Revolutionary War   moved from Moore's-Folly-on-the-Hudson to Mount Tirzah N. C. a beautiful spot in (now) Person Co. 'where the ax had never been laid to tree'. In 1779 he took command of a regiment of North Caroina State troops which participated in the first battle of Camden, in which battle the Americans under Gen. Horatio Gates were defeated by the British under Lord Cornwallis, and he was taken prisoner and carried to Charleston, S.C. Upon his release he retired to Mount Tirzah; after the peace he sold his estate on the Hudson to the U. S. Government. It is now the site of the U. S. Military Academy, (founded 1802).  Moore, Henshaw, supra. Mrs. Sophronia Moore Horner, Genealogist."SIX CENTURIES OF MOOR DE FAWLEY by David Moore Hall, printed 1904 by O.E. FLANHART Printing Co., Richmond, VA for himself..  page 38  [note there are errors in this book]

   see this web site by Jay Cross with wonderful graphics: The Moores of West Point  

Child of Stephen Moore and his first wife:
 1. Robert Moore 12 Nov 1769 Quebec - 27 Nov 1827 Person Co, NC
      married 11 Feb 1797 Sarah [Sallie] Bailey

Children of Stephen and Grizey Moore of Mt. Tirzah:
  1. John Moore 12 Nov 1769 Quebec - 7 Sept 1770
 2. Phillips Moore 14 July 1771 New York City - 2 Jan 1840 Person Co
      married 19 Jan 1794 Rebecca Moore
      married 25 Nov 1800 Elizabeth Dudley
 3. Frances Moore 5 Nov 1773 Westpoint, NY - Mar 1819 Person Co
     married in Caswell Co, NC 7 Nov 1791 Jesse Dickins ca 1767 Norfolk VA - 1836 Person Co NC
 4. Ann Moore 12 Jan 1777 Granville Co NC - 8 May 1864 Person Co  dsp
   "very amiable daughter Ann who has been a patient sufferer from early life and utterly helpless occasioned solely by wounding a finger with the point of a fine needle, and if living, she is still a cripple" noted in 1824 by John Moore Esq.
 5. Mary Moore 21 Sept 1778 Mt. Tirzah - 24 Apr 1851 Alamance Co NC
      married 11 Sep 1803 Hon. Richard Stanford 2 Mar 1767 - 9 Apr 1816
           he was first married to Jeanette dau of Gen. Alexander Mebane
 6. Marcus Moore 27 Nov 1780 Mt. Tirzah - 5 Sep 1797 Mt Tirzah dy
 7. Dr. Portius Moore 15 Oct 1784 Mt. Tirzah - 3 Dec 1847
      married 1st 1806 Frances Webb ca 1775 - bef 1811
      married 2nd in Henrico Co VA 1814 Lucy Wilson Pulliam ca 1790 VA -
 8. Cadmus Moore 30 June 1787 Mt. Tirzah - 4 May 1789  dy 
 9. Samuel Moore 15 June 1789 Mt. Tirzah - aft 1870 Person Co
       married 23 Nov 1810 Betsey Knott
10. Sidney Moore 15 Dec 1794 Mt. Tirzah - 29 Sep 1838
       married 13 Mar 1813 Mary Payne Reade

Grandchildren:

Children of Robert Moore and Sallie Bailey:
1. Yancey Moore 11 July 1799 Person Co NC - 4 Mar 1881 Carroll TN
     married Halifax Co VA 20 Dec 1827 Mary Ann Wade
      a. Sarah A Moore 1829 -
      b. Robert Yancey Moore 1831 -
      c. Richard B Moore 1833 - 1883
      d. William H Moore 1834 - 1860
      e. Eliza J Moore 1836 -
      f. John Bailey Moore 1838 -
      g. George Wade Moore 1839 - 1862
      h. Charles W Moore 1841 -
      i. Susan A Moore 1843 - 1874
      j. James Albert Moore 1844 - 1864
      k. Mary P Moore 1846 - 1873
      l. Martha P Moore 1847 - 1880
      m. Benjamin P Moore 1848 -
2. Albert Moore 1802 - 1851 Panola Co MS
    married Martha Ann Allen
      a. Sarah Ann Moore 10 June 1830 - 16 June 1860
           married 1852 James Jesse Smith  1823 - 1890 [2nd cousin]
           i. Dr. Henry Allen Smith 1858 - 1939
                       g grandfather of Steve Kelly
      b. Margaret Moore  
      c. Virginia Moore  
3. William Wesley Moore 1804 -
4. Gilbert Moore 1806 - 5 July 1883 Person Co NC
     married 19 Oct 1830 Lavinia Satterfield
       a. Frances Ann Moore
       b. Robert R Moore 1833 - 1862
       c. Baines William Moore 1835 -
       d. Thomas Barnette Moore 1837 - 1918
       e. George Lewis Moore 1839 - 1840
       f. George Washington Moore 1841 -
       g. Sarah Elizabeth Moore 1843 -
       h. Martha Jane Moore 1845 -
       i. Mary Celestia Moore 1848 -
5. Grizey Phillips Moore 1807 - 1840
     married Joel Sweaney
       a. John Robert Sweaney 1838 -
       b. Joel Phillips Sweaney 1839 -
       c. Sarah Elizabeth Sweaney 1840 - 1892
6. Elizabeth [Betsey] Moore 1808 - 1835
     married George W Ferrell
7. Ann Marie Moore 24 Dec 1810 - 6 Dec 1858 
             buried Moore Family Cem, Mt. Tirzah
     married William R Reade
8. Sarah Harriet/Henrietta Moore 1816 - 1864
     married Richard Henry Moore
       a. Sarah Jane Moore 1840 -
       b. Henry F Moore 1849 -
     married Person Co NC 14 Jan 1855 Anderson Thompson

Perhaps THIS is Dr. Moore's connection to the family -- it's his wife Julia's connection to the Mebanes!! Her aunt Mary Wood married William Mebane, a son of Gen. Alexander Mebane 1744-1795.

The Twentieth Century Biographical Dictionary of Notable Americans:
   Volume IX page 463
STANFORD, Richard, representative, was born in Dorchester county, near Vienna,
Md., March 2, 1767; son of Richard Stanford, and a descendant of Richard Stanford, who sailed in the ship Primrose from Gravesend, London, England, to Virginia, July 27, 1635. He removed from Maryland to North Carolina about 1793 and established an academy at Hawfields, Orange county, among his pupils being Thomas H. Benton, afterwards U.S. senator. He also conducted a farm at Hawfields and was married to Jeannette, daughter of Gen. Alexander Mebane, representative in the 3d congress (1792–95). She died in three years, and he married, secondly, Mary, daughter of Gen. Stephen Moore of Mt. Tirzah, Person county, N.C. (formerly of New York city), and granddaughter of Col. John Moore, a leading merchant in the early days of New York city. She died, Sept. 20, 1858, at Lambsville, N.C. He was a Democratic representative from North Carolina in the 5th-14th congresses, serving from May 15, 1797, to April 9, 1816, and was the first representative in congress to be called "The Father of the House." He was chairman of the committee on revisal and unfinished business during the 13th congress, and upon the assembling of the 14th congress was chosen chairman of the committee on rules and orders of the house. . . . Representative Stanford died at Washington, D.C., during the 14th session of congress, and was buried in the Congressional cemetery. The date of Representative Stanford's death is April 9, 1816.  [found at Ancestry.com]