As for Russia, the Fulbright has the final decision. Actually, they don't even give applicants a choice. They just choose what is best both for the candidate and the programme.
Hey, there!
I'm a Fulbright semi-finalist from Russia. My application is sent to 4 universities. But the Fulbright office is working on it, I have no information about chosen schools and details concerning my chances with each university. Whis is awful and stressful. I wish I could have more knowledge.