I can't comment on Sir John Ashton as I don't know much about him, but
for what it's worth I don't think it likely that he married a
Margaret, daughter of Sir John Byron.
The statement of marriage may have arisen by the mistaken
interpretation of the 1415 agreement whereby Sir John Byron in 1415
made a settlement of lands in Droylsden when his daughter Elizabeth
married Thomas, son of Sir John Ashton [VCH (Lancashire), iv, 284,
citing Byron Chartul. no. 1/23; no. 8/24]. The marriage is not
surprising, given that Byron had been a ward of Ashton, but that
wardship itself would seem to make the purported John Ashton-Margaret
Byron marriage even more unlikely.
As for Byron's one known daughter named Margaret, she married (1)
William Atherton, and (2) Sir Robert Harcourt.
The first marriage is attested by the papal registers. On 22 October
1436, in response to a petition by William de Athirton, knight,
nobleman, and "Margarete ^dilecti filii nobilis riri^ Johannis Burun
mulieris", it was decreed that their marriage, which had been
contracted in church, was legitimate [Cal. Pap. reg., viii, 577-78].
(This was required because William had previously committed
fornication with a woman related to Maragert within the third and
fourth degrees of kindred.) Note that they were apparently already
married by July 1435 [ibid., 570], although Margaret's name was not
then mentioned. As an aside, on 8 September 1440 John Byroun,
knight, entered into a lease for 10 years with William de Athirton,
son and heir of William de Athirton, knight, of a field in Athirton,
the rent being one rose yearly, if demanded. [Cal. Cl. Rolls, 19 H.
VI, 435], and this is the last known (by me) record of William
Athirton, though I have not done an exhaustive check of Athirton
records.
The second marriage is attested by the records of Ronton Priory, where
it is stated that "Roberto de Harecourt ... qui desponsavit Margeriam
filiam domini Johannis Berone militis" [Monasticon, vi(1), 258, cf.
Coll. Hist. Staff., iv, 272, which has "Margaretam" and "Johannis
Bevone, miliis"]. Ronton's account of the Harcourts is accurate and
independently verified (except for very early generations), and as
this was one of the last generations recorded (only two more followed)
it was much closer in time to when the record was made and the
marriage is more likely to have been known by those living at the
time. I'm not sure whether there is some other, more direct,
attestation to the marriage, but it is consistent with other records,
for the Byron and Harcourt families clearly had some association, as
well as a connection to Athirton and Ashton. For example, a fine of
16 March 1443 between John Byron, knight, and Robert Harecourt,
knight, and Margaret his wife, concerning lands in Athirton and
Assheton. (Presumably they acquired these lands through her first
marriage.) Robert and his wife Margaert (surname not given) were
married by July 1441 [Cal. Pap. Reg., ix, p.242].
Regards
Alan R Grey