Earlier I had noted the importance of DIFFERENCE is making humanity/human society what it is today. It follows that there is value in preserving differences. This is not meant to be racist, but it can be interpreted as such. Europe it seems to me needs people as labor - both upper end and low end - and as the means to support its social welfare system - someone must be around to pay into the system when the current population ages. But many of the people do not assimilate, and to the extent they do, the result is not the 'old ways', but some type of hybrid society (or pockets within the dominant society). You know all this.
I'm saying that Europe is not like the USA, Canada, nor Brasil, nor other South American countries, where the current society is already something different than before. In my opinion, Europe should restrict immigration far more closely than New World countries. Not for racial reasons but for cultural reasons. And, yes, cultural can include religion. There are likely cultural differences that should be preserved. Europe should not strive to be 'homogeneous' ('homogenizing') like the US.
The countries of sub-Saharan Africa, India, and China have internal growth that is needed, so that should be their focus. Every other country needs to consider their own situation, but they are likely like these, if not like Europe.
Edits above and here added: Wasn't that the reason for BREXIT? The cultural element, not the economics - except that with automation, every strata of society is affected. This is not to justify BREXIT, but Europe from my perspective would do far better maximizing trade, standardizing where appropriate to trade, protecting trademarks and product names, etc. and going easy on issues that affect the core dominant societies in each member country of the EU.
As to the need for immigrants to fuel the economies, the mega-trend is that automation in its various forms will soon, and for ever more, reduce the numbers of people required to sustain an economy. So, some of the current immigration value is quite definitely short-term benefit for long term pain.
One more comment here. Every country does well to invest in its people. A laissez faire approach to human development in an era of globalization is tremendously stupid. To the extent that Europe desires to maintain its culture, then these countries need to take to heart the imperative to maximize the human potential of their people. And also work to preserve the best elements of their culture and remembrance of its history.