In 212 AD Edict of Caracalla, granted Roman citizenship to all free men in Roman Empire. It increased the internal mobility inside the Empire. In 313 AD, the Emperor Constantine issued the Edict of Milan, which accepted Christianity.
We usually discuss about human migrations in language change context. But human migrations also change the culture and occasionally even the religion. We have two papers discussing Imperial Rome period and both shows the same event. In Imperial period there was large inflow from West Asia to the territory of modern Italy. The upcoming paper will confirm this and will pinpoint more precisely the most important source of those migrants. Anatolia.
""The demographic significance of Anatolia on a Mediterranean-wide scale is further documented by our finding that following the Roman conquest, the Anatolian population remained stable and became the geographic source for much of the ancestry of Imperial Rome itself.""
One simplistic explanation would be slave deportation but this do not explain why there is no similar shift toward Western Europe ( France, Spain ) or Balkanes which also were a source of slaves. Another more realistic explanation is that we are dealing with economic migration. Skilled craftsmen, paysants, traders, military recruits move to Italy just because they can earn more money there. Those new migrants will introduce new cults via religious preachers searching new adepts. Christianity, Mithraism , Judaism etc.