Please read J. H. Round "Peerage and Pedigree" vol. 2, pages 134 to 257, "The Great Carington Imposture," or at least through to p. 195.
https://archive.org/details/cu31924102029414
There are too many criticisms to list them all, even if we limit ourselves to the alleged link between Carrington and Smith.
Copinger thoroughly demolishes the document supposedly written by John Carrington, als. Smith, later "discovered" and accepted by the heralds, called "The Pedigree and Exploits in Foreign Countries of John Carington, Armiger, as related by himself to the year 1404."
The narrative of the Carrington document doesn't match known, established facts and dates that can be verified by available records. And it is written in such an amateurish way that even if the facts added up, the document wouldn't hold up as genuine.
A research fellow from Exeter (later Oxford), a Mr. Stevenson, analyzed the document for Round, and concluded that the language in the document is anachronistic (not of the time of the supposed author), inconsistent, and worst of all, includes "forms and usages that never existed." He opines that the author "may have read some Chaucer." He ultimately states that "the document bears the marks of spuriousness in every line."
It is difficult to conclude that the supposed narrative by John Carrington is anything but a forgery. And the entire Carrington case appears to rest on this document alone.
See p. 151:
"And on this document all depends : its opening words are the only evidence for Sir Michael's very existence ; its latter portion is the only evidence for the story that a John Carrington changed his name to Smith. The entire 'Carington' descent rests on this document; if its link breaks, the claim of the Smiths at once crumbles into dust.
My case is that the document is forged, and that the pedigree it contains is sheer invention, false at the beginning, false in the middle, and false at the end. And this I propose to prove by overwhelming evidence."
Round appears to have succeeded.
Central to our "quest": there is no proof that Sir Thomas Carrington had sons Edmund and John (John Carrington als. Smith, is alleged to have inherited the manor in Cheshire from older brother Edmund, and to have had it taken from him by his uncle George).
These are probably the most important points--see p. 171:
* The historians of Cheshire and the Cheshire records know nothing of Edmund or of John.
* Dr. Copinger does not produce record evidence for the existence of either.
* After Sir Thomas, who did not die until 1383, the next certain Lord of Carington is Sir George de Carington, who was undoubtedly in possession in 1402, and apparently so at least as early as 1397.
* Copinger claims that John succeeded his older brother Edmund, and that the manor was wrongfully seized by his "uncle" George.
* Then he claims that John returned to England, not as his brother's heir (George had two daughters), but as "he had byn made_executore_."
* Nor does the narrative know anything of his succeeding to the manor or of his uncle supplanting him.
Round shows, in excruciating detail, that Copinger took the forged document and made a bigger mess of things with more errors, exaggerations, amateurish bungling, and inconsistencies--again, he points out too many issues to list here. There are many details that are impossible (given the established facts and history), unproven, or unlikely (e.g. unlikely that John Carrington was sent to be raised by Neville, a busy general, when he was an infant).
It appears that our John Smith of Rivenhall could not have been John Carrington, son of Thomas Carington of Cheshire, could not have done what he is said to have done, and had many fictitious details added to his story.
Round also points to the similarities with the Croke-Blount story (pp. 191-194) as well as some jarring inconsistencies that call both narratives into question, and points out that the Croke and Smith families were founded by lawyers who rose in prominence under Henry VIII, became rich, and bought land. The pedigrees leading to Blount and Carrington appear during their grandchildren's generation, in the time of Elizabeth I, when many such false pedigrees appeared.
He also points out (p. 187) that the College of Heralds never approved the peacock arms.
I recommend taking some time to thoroughly read through "The Great Carington Imposture" before spending any more time on this project.
Full disclosure--I am from one of the Virginia lines, and would dearly love to prove the tale of our Carrington descent, but alas, it's a lost cause.