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Most of the Western Churches were already preaching, teaching and reading the New Testament in Latin by the time of Constantine. The Greek-speaking churches remained in the Eastern half of the Roman Empire.
Popular Editions or Recensions of the Greek text did take place, according to Jerome, at Alexandria under Hesychius, at Caesarea under Origen, and at Antioch by Lucian. But Jerome claims to have avoided all those popular local editions, and resorted only to "ancient Greek copies", which can only mean manuscripts at least 60-100 years older than himself (c. 392 A.D.) and hence contemporary with manuscripts perhaps 30 or more years older than Vaticanus and Sinaiticus.
It is possible that Jerome used manuscripts which reflected Origen's or Pamphylus' preferences. Or his chosen copies might reflect the type of text found in papyrus P45 or P66. But this is only a side-issue.
From the map, it can be seen just what groups of churches would be under the sway of the various editors and their recensions. The Alexandrian, Caesarean, and Antiochian areas all lay outside, on the periphery of the main bulk of the church communities spread across the Empire. Its not surprising that these independent communities had their own texts.
But it is highly implausible that the rest of the Empire would be significantly affected by their textual choices. The Latins would hardly embrace textual novelties, as the long resistance to Jerome's new Latin Vulgate demonstrates. In the process of assimilating Jerome's text, the Latins reversed many of his textual decisions, restoring the Latin text back to familiar readings.
The same would be expected from the central Greek Byzantine community, were a text like that of Lucian brought from the outside, i.e., Antioch. The Byzantines might indeed assimilate some of the readings of a popular preacher like Lucian, or Chrysostom, but any wholesale alterations would meet strong and vocal resistance, and a new text would only be adopted once older accepted and long familiar readings were restored to their traditional places. Lucian's text would have undergone a restoration process, conforming it to the proto- or early Byzantine text as recognized and venerated by the Greek-speaking Christian world.
This textual inertia would have resisted textual innovations and would have tended to preserve stability in the text-form copied by Greek scribes.