I don't think that sort of thing ever happened-- officially giving the imperium to a barbarian. Even at the end, when it was basically entirely controlled by barbarians, it would have been considered improper for Odoaker to become the emperor, which is why he (and his predecessors) supported puppets instead.
What's a barbarian? Aurelian was from Dacia. Diocletian was Illyrian. The Romans had assimilated many other people's before, but after Adrianople, the Goths and other Germanic tribes tended to become miniature kingdoms inside the Roman Empire and were never viewed as "Roman", even if (as in Stilicho's case), they were born of a Roman mother and lived their whole lives in the Empire. I think that was a big part of the problem.