The 'barbarianization' theory of the fall of the roman empire is often used in this context. To be fair, I think the theory is pretty old and goes back before the 20th century and the 20th centurys migration fears.
It was mainly popularised by Edward Gibbon, in his
Decline and Fall of the Roman Empire.
While the theory has gone out of fashion, I dont think it was specifically made to put across a political point. It just fitted well.
In 18th century Britain, the theory that barbarians had destroyed the Western Roman Empire suited the new ideology of imperialism - Britain saw itself as the cultural inheritor of the Roman Empire, with a duty to spread enlightenment and civilisation across a benighted, barbaric world. The fall of the Western Roman Empire was seen as a prime example of what could happen if barbarism was not proactively kept in check by the forces of civilisation. Basically, it was used as a justification for aggressive imperialism.
Nowadays, following the collapse of the European empires in the 20th century, the usefulness of this ideology has ended. We therefore now tend to see the collapse of the Western Roman Empire as having many causes, primarily internal ones, and that the pressure from the Germanic barbarian tribes was merely the straw which broke the camel's back.
However, some people now seem to be trying to resurrect this idea, but reinterpreted in terms of immigration and population migrations rather than actual full-scale invasions of entire tribes. I can see why they would want to do this, but it's intellectually rather dubious.
"Politics is the art of looking for trouble, finding it everywhere, diagnosing it incorrectly and applying the wrong remedies." - Marx (Groucho)