That depends on what you mean by 'barbarian migrations'. In my view, this pertains to the later Roman empire (end of the 4th century) when 'barbarian' tribes were migrating south to escape the coming of the Huns (migrating west from Central Asia/Russia).

A specific example that I am familiar with relates to the Goths who, by AD 375, were nearly annihilated by the Huns. Huge groups of Goths sought asylum in Roman land across the Danube and it was offered to them by emperor Valens. As to why Valens was keen to allow the Goths in, he was low on money and manpower (Julian's war in Persia a decade earlier had drained the empire of both and on top of that Valentinian/Valens had struggled with conflicts in Africa, on the Danube and in Britain). By allowing the Goths in to settle, Valens would see an injection of manpower (in the form of conscripted Goths) and money (in the form of taxes from the newly settled Gothic citizens). So, on the imperial side, there was certainly an eagerness to introduce barbarians into the empire to solve a spiraling problem of money/manpower.

However, as the Goths actually crossed into the Roman empire, the situation was badly handled by Valens' generals in the area. His officials and commanders mistreated the Goths, forcing them to pay extortionate amounts of money to allow them entry (not part of any deal with Valens). This culminated in the disastrous battle of Adrianople.

Though this was not a movement or a part of imperial policy, it does show that commanders on the Danube (who would have likely been local officers) were not keen on barbarian populations. They would not have done this if it was not a widely felt sentiment so we can certainly assume that there was a sensitivity on the northern borders of the empire towards barbarians and therefore it is likely that there were anti-immigration acts (though perhaps not to the extent which would constitute a 'movement'). Hope this helps.

The information above is provided in Ammianus Marcellinus' Res Gestae (books 30 and 31).

"Anti-Immigrant" sentiment was indeed present in Ancient Rome, especially later in Antiquity as various other groups were moving into the Empire, even if they were coming into Imperial service. But let's examine the state of the Empire and the various groups moving into the Empire at the end of the 4th and beginning of the 5th Centuries using the Goths as our case-study.

So first off, who were the Goths, how did they end up in Roman land, what was their relationship with the Empire?

According to Jordanes, the Goths traced their ancestry from Scandinavia and had settled around the Black Sea for some time, ending up on the Roman radar in the middle of the 3rd Century Crisis. They had led prosperous lives on the fringes of the Romans world until the depredations of the Huns forced them to flee their ravages and try to escape to the safety of Roman territory. The Goths split into Visigoths and Ostrogoths at this time, the Ostrogoths remaining under Hunnic authority for a time while the Visigoths crossed into Roman territory.

Traditionally, the Goths are marked as the epitome of the participants in the Voelkewanderung. An entire society uprooted by the onslaught of the Steppe peoples, who then poured over the Roman borders in their thousands bringing their entire populations in search for a new home. The Romans let in some of the refugees but not all, eventually enmities between the groups flaired up and the Goths were responsible for some of the worst disasters in Roman history, the battle of Adrianople and the 410 sack of Rome come to mind. Eventually the Visigoths settled in Aquitaine and Iberia and the Ostrogoths took over Italy.

From this traditional view we get a lot of the misconceptions that come down to the modern day.

This view has a LOT of flaws in it.

Any attempts to reconstruct the migratory past of a pre-literate society are always going to run into some large issues. There is archaeological evidence of a (partly) Germanic culture occupying space around the Black Sea that seems to have fallen apart alongside the rise of the Huns, the Chernyakhov culture. However any attempt to solidly connect this culture to the Goths is going to be tenuous. Efforts to construct a migratory history of the Goths based almost solely off of Roman accounts, which were often written well outside any realm of contemporary, are doomed to failure.

However the biggest issue with this traditional view, and this gets to the gist of your question, is that the Goths were perfectly willing to not only work alongside but fight for the Romans. They did not arrive on the Danube river looking for Roman blood and maraud the countryside until they got their fill of it. Indeed it is hotly debated just when (if ever) Gothic identity solidified around "visigoths" and "ostrogoths". Peter Heather argues that this process did not happen until the various groups that all comprised the forces under Alaric were in Roman territory.

Throughout their interactions with the Romans there were Gothic forces who clashed with the Romans then turned around and fought alongside them, only to then end up back at war with the Romans. The "visigoths" fit this to a T.

So the Romans and Goths for example fought alongside and against each other at various times, and this relationship was by no means unusual. Huns, Alans, Suebians, Vandals, Goths, and so on all served in the Roman army at various times as well as fighting them.

So with all of that what can we say about how the Romans viewed these people?

Naturally, it is quite complex and many barbarian new comers would attain incredibly high status in the Empire,for example Stilicho ,a Roman of Vandal extraction, even ruled the western half of the Empire at one point. Indeed other figures such as Alaric coveted Roman office and positions of authority within the Empire. But the Romans were not always so accommodating.

Stilicho was executed and in the aftermath of his death a wave of what were essentially pogroms against "barbarians" within the empire happened and large numbers of these people were killed or expelled from their communities (many survivors would join Alaric and this is what inspired his march on Rome). The Romans were also quite content to allow the barbarian populations within their borders suffer in times of hardship as a way to weaken any challenges to Roman power. The Romans famously abused the Goths who had come fleeing the Huns and left them to starve when things got tough.

But this attitude was not always the MO of Romans. Many post-Roman polities that still identified as Roman were quite keen to placate their new barbarian overlords and this went both ways. Many barbarian rulers held a great deal of respect for Roman practices and ideas, and went to great lengths to preserve them as much as possible.