Well if you ask for Antonov that range starts from Antonov 2 Prop to Antonov 225...
Clear answer is Yes, there are "non-Russians" flying Antonovs.
There are plenty of Antonov, Tupolev and Ilyushin aircraft operated in each of the former Soviet countries and they have local crew flying those (russian speaking but technically not Russian...).
Additionally there are modern versions of Russian metal such as Antonov 148/158 which were sold to Cuba and North Korea for example and I am sure these aircraft are operated by local crews (there was a report recently that the first new Antonov for Cubana was delievered with a mixed Russian and local crew).
China and Indian Air Forces operate Antonov aircraft so there must be rated local guys out there...
Cheers