In the early days of the empire, Rome was big, probably around 1 million. There were some number of Jews (thousands at least, perhaps as many as 40,000 by some estimates, although that’s probably high). There were also other foreigners in Rome – probably mostly Greeks, but also some Gauls and Syrians and such. There were also many slaves in Italy, perhaps a third of the population.
Mostly those slaves were from Europe, obtained in wars of expansion: Gaul, Hispania, Germany, Britannia, Greece, etc. Wiki suggests that their European origins is why they didn’t have much affect on Italian genetics because they were European, but that’s wrong. You can certainly detect genetic differences between Gauls and Italian, Germans and Italians, etc. Wiki is correct in saying that you don’t see much sign of this ancient immigration in Italian genetics, but it’s not because they were just like Italians: it has to be because they died out.
The cities were population sinks, and collapsed with the Empire. I doubt if slaves had high birth rates: certainly those working in mines or quarries didn’t. Nor did gladiators. One way or another, the foreigners in Italy, the vast majority of them, disappeared.
Now you see some signs of other stuff in Sicily or Calabria today, but that seems to be later, from Arab or Byzantine times.