They are not going to - even AMD's optimistic estimates talk about 40% improvement. This is way too little to close the performance gap. At best they can match top speed Intel CPUs from ~2011 (late Sandy Bridge).
Zen is (according to AMD, so I guess it is optimistic) is supposed to bring 40% improvement in instructions per clock. That would put it around Sandy Bridge level. They would have to pull off 100% improvement to be competitive at high levels once again.
It is not a problem of Spaniard moving to Germany. It is a problem of Romania going bankrupt in 2 years when having to pay CBI at the same level as Germany. Or paying it at vastly lower level which means that half of Romania moves to Germany since they will get more for sitting on their asses and average salary in Romania. Third way is for western countries to increase subsidies for east by huge amount and I don't think it is going to fly, especially not with upcoming referendum about UK exiting EU. The basic problem is that EU is very unequal economically and this is not going to change anytime soon.
So: immigrants move in, take the low-level jobs left by natives (who decide they are better off on UBS than being paid peanuts for flipping burgers), work 2 years (not a hardship since they are still paid several times more than at home) and finally move to UBS. That's not a solution, but a small delay.
The limited housing supply affects everybody equally - there is no priority list where all citizens from given country are assigned housing first and only then immigrants. So mass influx of immigrants would actually make life miserable for everybody. Take UK for example: housing situation is quite bad and still net immigration is huge.