Heh... of course, that's not a good analogy, as the Romans tried their darnedest to keep their borders closed. Rome should have lef Carausius rule Britain, and made alliance with his independent state. It would have simplified the border problems and solidified the NW provinces right when they needed it. Ah well. Instead they probably induced Allectus to assassinate and supplant Carausius, and he wasn't effective at any other activities, as it turned out, just as the Emperor in Rome wanted. Britain was brought back into the empire, Allectus was executed, and the province was divided into four or five parts, with separate governors. Of course, since much of my ancestry is from there, I'm glad it worked out that way, else, wouldn't be here, and probably no one reading this would be, either.
They expanded the borders and incorporated the foreigners not just into their populations but their armies as well; in the end the Roman Empire collapsed because too few of its people were “Roman”.