At some point it will become intolerable and the citizenry will rise up and started killing the invaders. That’s what happens when the govt just ignores its own people long enough.
What is most likely to bring that on will be the "refugee's" near certain violent response to any attempts by EU states to reestablish control. I'd bet that is why they are so hesitant to take action - they (EU) know that they've let the serpent in and it won't leave willingly.