It's important to preface this with the fact that the Romans didn't really have a concept of 'race' or 'nationality', and while there was prejudice against barbari (essentially, non-Latinate foreigners) 'racism' didn't exist.

This is also somewhat the case for borders. The edges of the Empire were termed limes (lit. 'boundary' or 'limit'). These consisted mostly of rivers and occasionally walls, and while it was important that the Emperor had control over these borders, there was no real fortification there, and the borders were more dynamic than the static lines we think of today. Borders were meant to be crossed - Rome had extensive trade with outside civilisations, most notably India and Meroitic Kush for spices and perfumes. Most prominently in the earlier years in times of expansion, Rome didn't really have issues with 'immigration' - they were set on expanding the Empire and its population as much as possible. For the most part, it was the Romans that were the unwelcome ones invading other regions! ;)

As for notable individuals/events: in AD 212, Caracalla issued an edict (1.5.17) that made all free men in the Empire's bounds were to be given full Roman citizenship and ensured provincials had the same rights as Romans. It's thought that he did this for the purpose of increasing tax or the number of people available to the army, but in doing so he effectively put an end to the idea that provincials were somehow inferior.

Also, later in the fourth century there were small 'border-control' type armies trained specifically for the frontiers in case of attacks on the border. From my limited knowledge of the time (I specialise more in 1st-2nd centuries AD) they were a bit rubbish.

The debate on Roman migration/fall of the Empire is pretty alive right now and I'm likely not the most knowledgeable person on this sub for it, but it's important to know that the argument that the western Empire was ruined by "hordes" of barbarians is simply untrue. I won't argue that migration had no effect whatsoever, but the Empire's fall was the result of many and complex factors including economic troubles, the gap between West and East (Byzantium), and corruption in the government.

Hope this helped somewhat!

I can speak mostly to Rome's dealing with the Germanic "barbarians" in the fourth and fifth centuries, which I think gives a decent answer to your question.

One key thing to note is that Rome's immigration policy, if we can call it that, was largely an extension of its wider strategic frontier policy. Rome would regularly play factions against one another as a way of maintaining a manageable balance of power and keeping border peoples friendly to the Empire. Instances like the Marcomannic War represent a temporary breakdown of this system. In general, however, Rome liked to keep its closet neighbors friendly with gifts, and these groups desperately wanted access to Roman goods (usually because they served as status items and helped leaders assert power).

This system extended to immigration. For example, when two groups of Goths, the Thervingi and Greuthungi, asked for asylum in Roman territory in 376 (they were reportedly fleeing the Huns), Valens only admitted a group of the Thervingi. Peter Heather argues that this was a way of keeping the Goths divided. (The Greuthengi managed to get into the Empire, however, but for now, that's beside the point.) Valens likely admitted the Thervingi because he regarded them as a source of military power. Indeed, part of the deal for the Thervinigi was that they provide military service in the war against Sassanid Persia.

In the late fourth century, Rome was still strong enough to pick and choose (to some extent) what groups entered the Empire. As it turns out, however, managing a large group of Goths in Roman territory proved difficult, and the Goths eventually rebelled, defeating and killing Valens at Adrianople in 378. This group of Goths, after they sacked Rome in 410, were eventually settled in Aquitaine as Federates (a sort of self-governing Roman ally on Roman territory).

As Roman power waned in the early fifth century, other groups entered the Empire without formal sanction. The Vandals, Seuves, and Alans crossed the Rhine River in 406 and then moved through Gaul and into Spain. Unlike the case of the Goths, these groups were never admitted into the Empire, and Rome opposed their movements (albeit with little success). In fact, the Visigoths fought against Vandals and Seuves in Spain on behalf of Rome (though as the Empire declined, the Goths began pursuing their own agenda and incorporated this territory into an emerging Visigothic kingdom).

The Franks are less clear case. There is some evidence that they were given Federate status (similar to the Visigoths) and that they had some sort of relationship with a Roman general in Gaul, but the records aren't entirely clear. Regardless, they moved into former Roman territory when Rome's grip on the area was very weak.

Throughout the late fourth and fifth centuries, Rome relied more and more on Germanic peoples in the army. Many Roman generals, specifically Stilicho and later Odoacer, were Germanic generals of considerable influence in the Roman army. This "barbarization" of the Roman army is the subject of some debate, but as Roman power faded in the fifth century, the state became increasingly reliant on Germanic allies. At the Battle of the Catalaunian Plains in 451, for example, the Roman general Aetius defeated the Huns with a large allied force consisting of Visigoths, Franks, Burgundians, and Alans. Ironically, Aetius was also well known for using Hunnic allies in his wars against other Germanic groups in Gaul.

To summarize, then, in the fourth century, Rome was still powerful enough to determine who entered the empire. Its policy when dealing with immigrants reflected a long tradition of supporting border allies and keeping barbarian groups disunited. Once Roman power began to fade in the fifth century, Rome lost its ability to determine what groups entered the empire. Its army became increasingly reliant on Germanic allies, and Germanic generals gained more and more influence in Western Roman politics. This process culminated when Odoacer, a Germanic Roman general, deposed the last Western Roman Empire, Romulus Augustulus, in 476.

Sources: Heather, The Fall of the Roman Empire; Heather, The Goths; Heather, Empires and Barbarians; Halsall, Barbarian Migrations and the Roman West; Goffart, Barbarians and Romans; Goffart, Barbarian Tides.