As was common for pre-Reformation music in England in Latin, the large majority of manuscripts were destroyed during the Dissolution of the Monasteries by Henry VIII (and a large proportion of English-language sacred music was destroyed during the subsequent reign of Mary, during her attempt to re-impose Roman Catholicism on the island).
With the fall of Cardinal Wolsey in 1529 the college founded by him was not able to devote resources to such music and so the manuscript was discontinued, and this situation was probably the reason for Taverner's departure that year.
The other Mass-setting (Missa Ave Maria, also for 6 voices, which is the finer of the two and an outstanding work with similarities to Taverner's Missa Gloria tibi Trinitas) was copied into the partbooks mid-century along with five other settings by other composers, though the dates of composition of both Ashwell Masses were considerably earlier: their style indicates dates of composition possibly even before his appointment at Durham.
A personal connection with Ashwell would account for the inclusion of his Masses in the Forrest-Heyther Partbooks, copied either by Taverner or for him when he became head of music at Cardinal College, Oxford in 1526.
Ashwell's reputation survived at least until the end of the 16th century, since Thomas Morley listed him as an authority in his 1597 treatise A Plaine and Easie Introduction to Practicall Musicke.