We can use Lyon as an example, if specification's needed. Additionally, what benefits, if any, would Roman Citizenship have confered outside of Italy after the collapse of Roman rule in the West?
Well, after the Antonine Constitution of 212 CE, the free population of the Empire was legally Roman. There were still subdivisions (honestiores and humiliores) that had different sets of rights, but it really wouldn't make sense to advocate for Roman citizenship after that. Of course that couldn't be enforced and plenty of people wouldn't have cared about Roman law anyway.
Ulpian and Cassius Dio are the guys to go to about this. Ulpian moreso, I think.
About the end of the Empire in the west stuff, that's not my territory. 212 CE is already pushing the limits.