According to the book
Foreigners at Rome: Citizens and Strangers by David Noy (London: Duckworth, 2000), the main sources of foreigners in Ancient Rome were Gaul and Hispania, Central and Eastern Europe, Greece, Asia Minor, Syria, Egypt, North Africa and Jews (p. 205 ff.). However, it seems that most of those from outside of Europe were ethnically Greek, while Jews never fully assimilated into Roman society, and that the city's foreign population at its peak was only ~5%. Also, foreigners (especially slaves) had higher mortality rates and lower birth rates than natives and were sometimes subjected to mass expulsions. Here are some relevant passages from the book:
Population Size
Immigration was essential to Rome both demographically, to increase or at least maintain the size of the city's population, and socially, to provide skilled workers and soldiers. The slave trade met some of the requirements, but free immigrants were always needed. Provincials probably began to outnumber Italians among newcomers to Rome in the first centuries BC and AD. The third century AD, when all recruitment for the Praetorian Guard was done in the provinces, may have seen the numerical peak of Rome's foreign population. It is plausible to suppose that at least 5% of the city's inhabitants were born outside Italy in that period; the reality could be much greater.
Expulsions
Foreigners who did not have Roman citizenship were always liable to summary expulsion from the city, and by the fourth century the possession of citizenship was no longer protection against such treatment. Although there was a certain amount of xenophobia within the Roman literary class, expulsion was only used in certain circumstances: to deal with the actual or potential misdeeds or alleged bad influence of specific groups (which could be defined by nationality, religion or occupation), or to counteract the effect of food shortages by reducing the number of mouths to feed. Expulsions were probably not carried out very efficiently, and were always short-lived.
Mortality Rate
It is generally agreed that mortality was probably higher in Rome than elsewhere in the Roman world, because of insanitary living conditions and the risk of contagious diseases; diseases such as tuberculosis may have been endemic. Newcomers to [17th-18th c.] London were more susceptible to plague than natives were, and the same point has been made about the greater susceptibility of Rome's immigrants to plasmodium falciparum malaria. Tuberculosis might be particularly dangerous to the young adults who probably formed most of the immigrant population. [...] Slaves are likely to have suffered from higher mortality than the free population, and immigrant slaves would have been particularly vulnerable to diseases which were not prevalent in their homelands.
Birth Rate
It is also likely that the birth rate would have been lower at Rome than elsewhere. Many migrants coming to the city would already have spent some of their fertile years elsewhere, and the slave part of the population would have been less fertile than the rest. Free male citizen immigrants may have postponed marriage until they had access to the corn dole, which from the time of Augustus was only available to a restricted number of recipients. In London, for similar reasons, the natives were closer to reproducing themselves than migrants were, and the same would almost certainly have been true for Rome.
Asia Minor
Although literature emphasizes the significance of Asian slaves at Rome, inscriptions present a rather different picture. The large number of epitaphs in Greek, especially for people from the province of Asia, is consistent with the large number of recorded peregrini [foreigners] in suggesting that the migration of people of free status was particularly significant for this area. The evidence is, however, almost exclusively concerned with the Greek population of Asia Minor, and there is very little sign of people of non-Greek background coming to Rome except as slaves. This is consistent with the general predominance of the most romanized/hellenized section of their home society among free migrants to Rome.
Syria
However, most Syrians arrived at Rome through the workings of the slave trade. Syrus was a common slave name, although not necessarily given only to Syrians, since the association Syrian = slave seems to have been very widespread....
"Voluntary migration from Syria to Rome would probably have begun in the late Republic. Most of the evidence, however, is from the second century AD or later. There is a clear implication that some of the slaves and ex-slaves labelled Syrians in the literary sources were thoroughly imbued with Greek culture, whether their ancestry was Syrian, Greek or mixed. Solin (1983, 722) notes that Syrian immigrants in general tended to be of Greek descent or at least to be from the most hellenized part of Syrian society.
Egypt
Most references to Egyptians at Rome concern Alexandrians, apparently of Greek extraction, rather than 'indigenous' Egyptians. On the other hand, the stereotyped Roman image of Egyptians concentrated on the aspects of their behaviour perceived as most outlandish, particularly the worship of animal-gods, and largely ignored the Greek component of their culture. There seems to be something of a contradiction between image and reality which may be due at least in part to anti-Cleopatra propaganda and its legacy.
North Africa
North Africa contained some cities which were Greek, Libyan or Phoenician foundations, but many of the main population centres began as Roman colonies (notably the re-established Carthage) or military settlements. Ricci (1994b, 198) believes that the colonization programme of Julius Caesar and Augustus in North Africa also stimulated a population flow from there to Rome. The inhabitants of the area came from a wide variety of ethnic backgrounds (Italian, Greek, Punic, Libyan, Berber, Jewish), but, as with other areas, it is likely to have been the most romanized/hellenized section of the population which provided most of the free migrants to Rome.
Jews
The group which made the greatest effort to retain a separate identity was the Jews. In their religious and communal institutions, their use of separate catacombs, their epigraphic and liturgical use of Greek, and even their naming practices, they behaved differently from others and were able to pass on a Jewish identity, so that people whose ancestors had lived at Rome for generations and who were otherwise well integrated into Roman society were still identifiably Jewish.