Yep. Rome was probably the most cosmopolitan empires of all time, largely because in comparison to other empires Rome would grant to conquered people or people who allied with the Roman state. For instance Caesars most crack units were made up of Gallic people's who he extended citizenship when he invaded Gaul (modern France)
The Romans argued about immigration just like we did.
In 48 AD, a discussion took place in the Roman Senate concerning the admittance of members of the Gallic aristocracy to the venerable body.
According to the Roman senator and historian
Tacitus, there was opposition to the move; some senators said that Italy was perfectly capable of providing its own members, and that it was enough that northern Italians had been admitted without having to resort to foreigners who had been, until recently, their enemies in war.
But as Tacitus reports it, the then-emperor Claudius
championed the move:
My ancestors … encourage me to govern by the same policy of transferring to this city all conspicuous merit, wherever found. And indeed I know, as facts, that the Julii came from Alba, the Coruncanii from Camerium, the Porcii from Tusculum, and not to inquire too minutely into the past, that new members have been brought into the Senate from Etruria and Lucania and the whole of Italy, that Italy itself was at last extended to the Alps, to the end that not only single persons but entire countries and tribes might be united under our name.
We had unshaken peace at home; we prospered in all our foreign relations, in the days when Italy beyond the Po was admitted to share our citizenship…. Are we sorry that the Balbi came to us from Spain, and other men not less illustrious from Narbon Gaul? Their descendants are still among us, and do not yield to us in patriotism.
Everything, Senators, which we now hold to be of the highest antiquity, was once new.
What the Romans can teach us on immigration and integration
The difference between Sparta and Rome was Sparta never changed. They fought the same, used the same equipment, had the same style of government no matter who they fought. Persians, Athenians, Thebans, Macedonians whatever they fought the same and expected their training to win the day. Until they were figured out, started losing fights, their population started to decline because of low birth rates, they were stuck in cultural stagnation which is death for any society.
The Romans used a Spanish sword (gladius), their shield and battle formation a Samnite design. When they were getting beaten by Carthage at sea they copied Carthagenian ship design. The Romans didn't like cavalry all that much so they hired Numidians and Gauls who were good at it.
Point being Rome integrated other cultures and people into their system and made them Roman, without that openness to societal change Rome wouldn't have lasted. They would've been a flash in the pan like the Spartans.
Hell, their emperors were coming from Africa after awhile
Septimius Severus - Wikipedia