My name is Dov Oppenheim of Israel.
My great grandfather from my mother side was Zvi Kushnir from Ukraine. He was born on 1887, his fathers name was Efraim Kushnir.
As far as I know he had 2 sisters that moved to Argentina with him at 1910.
I have no idea of other family left in the Ukraine, I do know that Kushnir is a very common name there.
Your great grandmothers name Betya(Basya) Chaimovna Kushnir suggest that her fathers name was Chaim.
I will try to look at records from Polonnoye area.