My thoughts are that the flow will keep coming in ever increasing numbers although the winter will slow it down. Then eventually after a few years of this happening the housing prices (and therefore rent) will skyrocket as the ones that don't have children will know that if they have a child they will automatically get to the top of the list especially if they have more. The burden of housing benefits which are now inflated coupled with cash benefit payments (many are uneducated/don't speak the local language/have no skills that are useful) most are unemployable and therefore don't pay taxes. Education places, health places (doctors and hospitals) will not be sustainable in the short term and the system will collapse or at least become intolerable as numbers rise.
This could end up a civil war in a short period unless Europe can keep a lid on it. I don't think it can be done personally, unless the flow is slowed right down.
Civil war is a term best used elsewhere, what is most likely is that fringe parties become the norm and the EU grinds to a halt.
After that I wouldnt be surprised at a few pogroms.