Where can one find lists or anything similar of monks of the Ordo
Sancti Basilii Magni Rutheniae (OSBM) in the 17th century? The OSBM is
a Uniate (i.e. Greek Catholic) order of orthodox priests who had
accepted the church union of Brest. Are there any registers of
Basilians who entered the order around 1620-1650 left? If so, where
are they now?

If such lists have not survived and the information is lost, an
explanation of the reasons for this would be welcome as well.

I have already addressed this question to several institutions of the
OSBM in Poland, Ukraine and Rome without receiving any reply.

Be careful about not confusing the Ruthenian (i.e. East Slavic) OSBM
with one of the three other Basilian orders in other countries. They
are not related, and the information is needed for an East Slavic
inhabitant of the Polish-Lithuanian Commonwealth.