At what point in time did the pronunciation of the Greek β change from “B” to “V”?

Looking at Sidney Allen’s Vox Graeca, we know that Plato (Cratylus 427a) describes both δ and τ as stops. The first unequivocal evidence is the differentiation between б and в in Cyrillic in the 9th century AD. It turns out though that at the same time, beta was being transliterated in Georgian as as ბ b rather than ვ v. Cicero (Fam ix 22.3) says that βινεῖ ‘he fucks’ is pronounced as Latin bini. On the other hand, Allen concedes that some non-Attic dialects (Boeotian, Elean, Pamphylian) may have started fricating voiced stops as early as the 4th century B.C.

Allen is reluctant to commit to any time of transition. The really long description at Koine Greek phonology – Wikipedia, pitting Allen against Gignac’s investigation of papyri, indicates that it was probably over a drawn-out period, with variation by region or register. However, the usual assumption I have seen is frication starting as [β], around the 1st century AD.