~3.5% of the US population is Illigal immigrants. If they want to come here they have. In fact over the last five years more people have moved from US to Mexico than Mexico to the US.
It's not really an issue because our 11 million illegal immigrants don't just desired to move to Montana they do what other people do and arrive in waves. Only issue US has with internal migration has to do with Job openings which are fairly easily to predict if hard to deal with. EX: Boom towns from fracking.
And then those people would repatriate funds, raising the quality of life in poorer countries and reducing the need to emigrate.
Labour is a tradable good, and as such the labour market must be an open as for any other goods (or services).
My point is this. In the USA, there are less cultural and economic differences between the states, so there won't be as many mass immigrations from one state to another. Now think about if USA decided over night, that it would open it's southern borders to all of South America, I think we all know what would happen.
And the same thing could happen in EU, because you get to huge cultural and economic differences on the borders of EU. Just think about Turkey and negotiations (their demands) to become part of EU. That would be a total disaster.
What good is EU or USA, if it can't keep ILLEGAL immigrants away ?