A lot of people seem to be under the impression that the use of a particular coat of arms by a Colonial family is "proof" that they were descendants of that family back in England (or occasionally elsewhere).
Not necessarily.
The British College of Arms governs the use of arms by persons *in Great Britain*, but they never were too particular about what a bunch of upstart Colonials did or didn't do (unless said Colonials contacted them directly - which a few did).
There is at least one well-known case of "assumed arms" (alias "arms of pretension") in Colonial history: the Dorseys of Maryland, who arbitrarily assumed the D'Arcy arms on the grounds that the name was so similar they must be ancestors. Nobody bothered about this...until Y-DNA testing in the late 20th and early 21st centuries upset the applecart and proved conclusively that the Dorseys were not only *not* "lost" D'Arcys, they were Irish from way back, probably originally O'Dorchaidhes. However, since there *are no rules* governing the use of coats of arms in America, they can go right on using those arms just because they like the design.
The Washington arms, now used by the District of Washington, DC, are an intermediate case. The Colonial Washington family probably *were* descendants of the Wessingtons (Washingtons) of Washington Old Hall, Durham (now Tyne and Wear), England. But exactly how they were connected, and whether they could use the arms undifferenced, is undetermined. Again, nobody bothered about it...and with the arms now territorial rather than personal, nobody is going to.
A few prominent families wanted to put their right to a coat of arms totally beyond question, and contacted the College of Arms directly. The Scarburghs of Accomack County, Virginia, had never been armigerous, but they wanted to be, so they paid the fees to have arms drawn up specifically for them and their descendants. The Famous Lees of Virginia wanted to stake their claim to descent from Lee of Coton, and had the assistance of an ally inside the College of Arms (John Gibbon, Bluemantle Pursuivant, who had worked for Col. Richard Lee "the Immigrant" in his younger days) - but even so, what was authorized for them was Lee of Coton quartered with Astley of Nordley, based on the family tradition of marriage with an Astley heiress. (Eyton, in his "Antiquties of Shropshire", eventually unearthed the precise transmission of property.) But descendants - or, especially, antecedents - of the Coton Hall Lees in England would use the Lee device unquartered.
As far as is known, only one Colonial resident had an undeniable claim, not only to arms, but to membership in the lesser nobility, that being Thomas Fairfax, sixth Baron Fairfax of Cameron, who came to Virginia as a young man to see exactly what he had inherited from his mother's family (the Culpepers), liked what he saw, and settled down and stayed. (He hired a young surveyor named George Washington - see above - to mark out the boundaries of exactly what he owned.) But Lord Fairfax had no children (it is said he went to America in the first place because of disappointment in love), and the later history of Fairfax of Cameron is, well, complicated.