So is the religion the biggest problem or just a symptom?

I'd agree immigration must have limits and don't recall any prominent voices calling for unlimited migration or even limitless compassion.

Being a union, one can hope the load of migrants fleeing wars and violence would be spread among all the states, in proportion to their ability. Economic migration is a more complex matter since everyone benefits from it up to some threshold. Where it crosses into an undeniable problem is subjective. If it negatively affects me personally then it's bad, especially if I can overlook all the benefits I accrued (like lower prices of commodities and services).