The Romans had by the time of the official "fall" of the Western empire already introduced the practice of stationing Germanic mercenary cavalry directly with landowners and towns in Italy, so that the mercenaries could collect their own wages directly as taxes. This feudal societal organization would basically remain unchanged in the Ostrogothic and Lombard kingdoms. Landowners and mercenaries had a shared interest in preventing the peasants/commoners under their control from leaving if that impacted income, and would be definitely capable of hunting them down if they did.

I do not have the impression, if you look at the sources covering later attempts by the Eastern Roman empire to expand their influence in Italy at the expense of the barbarians, that they had much popular support for doing so. On the contrary: small Lombard feudal armies for instance regularly defeated larger but very low morale locally sourced (Eastern) Roman armies. And parents complained about their children dressing as barbarians to look cool. That doesn't give the impression that the average former Roman was willing to risk his life to be able to live under an emperor in Italy. At best you could describe it as an attitude of apathy.