So now he's "Sir" John Greene and "Lord of Drayton", which leads me for one to suspect that he's been mash-merged with one or more members of the Midlands Greene family.
ENOUGH!!!
Of the "14" sources listed on the Source tab, ONLY TWO are accessible without a MyHeritage subscription (which means the rest are say-so family trees).
The "Visitations of Bedfordshire" show a Bedfordshire Greene with *NO CITED PARENTS*. It has been proven that this is *not* "John Greene the Fugitive".
The other accessible source is Burke's Peerage, discussing the Latimers (not the Greenes) and is rather obviously dependent upon the above Visitations - with the same problem.
In plain words, the "sources" don't prove a blasted thing.
John "the Fugitive" Greene is of unknown ancestry, and has been MIS-grafted onto the Greenes of Drayton in an attempt to give him some importance beyond being someone that Henry VII had it in for.
He may not even have been a native of Gillingham, Dorset, as it seems his connection is that he bought an estate in that town.
There is no "hard" evidence for any Mallorys remaining in Dorset by the 15th century.
The only known Greene-Mallory connection was several generations earlier, and in Leicestershire and Northamptonshire - NOT Dorset. (See transmission of the manor of Sudborough, Northamptonshire, and note that the Mallorys kept trying to grab it back - which is how we know the details.)
Shaw's "Knights" (vol. II p. 30) has a John Greene knighted on (or shortly after) June 17, 1497, either at the battle of Blackheath or "at the Bridge Foot on the King's entering London" following the battle. (There is inconsistency in the existing sources.) He is noted, though, as being "of Essex" - not Dorset, and not Northamptonshire either.
The connection of Greenes with Gillingham, Dorset, apparently begins with Robert Greene, Gent. (c. 1490-c.1558), who purchased an estate called Bowridge Fall in that vicinity, probably early in the 16th century. He is thought, but not proven, to have been of a cadet line of the Greenes of Northamptonshire; but whether he had any prior connection with Dorsetshire is unknown. Neither is it known whether he was in any way connected with Sir John Greene "of Essex".
As to the known Mallory-Greene connection, it runs like this:
Ala Mallory, daughter of one Sir Anketil Mallory and sister of another, married a Thomas Greene "of Isham" in Northamptonshire sometime toward the end of the 14th century. Her father, or brother (it isn't clear which), gifted the couple with a manor in Sudborough, Northamptonshire, probably as a dowry. Ala and Thomas had a son John, who married someone named Isabel; but Thomas died relatively young, possibly while John was still a minor, and Sir William Mallory (of Papworth St Agnes), son of the last Sir Anketil, was not slow to take back possession of Sudborough. John died before 1445, without regaining possession, and his widow Isabel and son Thomas took up the fight after Sir William died in 1445. Isabel apparently held possession for a time, and then Thomas Mallory, son of Sir William, got the manor away again. But Thomas Green eventually recovered possession, probably after Thomas Mallory died in 1469. Thereafter the descent of the manor is harder to trace, because the legal wranglings were at an end.
Oh, in case anyone hasn't noticed: Thomas Mallory of Papworth St. Agnes was at one time considered a strong candidate for authorship of the Morte d'Arthur; but it appears that he was never knighted (and that he paid a fine to *get out of* being knighted), which puts him out of the running. He may actually have wanted to avoid being confused with the infamous Sir Thomas Malory of Newbold Revel (who very probably *was* the author), who had a disturbingly long rap sheet and multiple imprisonments (plus several jailbreaks).
Exactly who Thomas Greene "of Isham" was is not clear, but it's probable that he was not of the major line of Greenes - the Mallorys would not have been able to shell-and-pea King's Justiciars and the like.