I'd love to hear some examples where this has caused technological stagnation and blocked skilled migration -- US and Canada provide solid counter-examples to these claims. Both countries take in immigrants at both ends of the economic spectrum, from coveted H1B high-paying tech jobs, to seasonal farm workers who exist in slavery-like circumstances. Both countries show significant, sustained technological innovation.
I agree this is destructive to the social fabric, and it separates us into strata. But I have a hard time believing that immigration is the cause; a simpler explanation would interpret it as one of the symptoms, with the underlying causes being closely related to the relentless transfer of wealth to an ever-shrinking "elite".
A better example might be the low price of labour-intensive hand-picked fruit (particularly berries) in the US and the correspondingly low agricultural automation. It's not that machines don't exist, or that berry picking is going to prevent the next Facebook, it's that they aren't cost effective in the short to mid term even when longer term investment would eventually improve the whole industry. This is really hard to measure and I don't have any good sources for either side of the argument right now.
I also want to make a distinction between skilled and unskilled migration. I don't think skilled migration is harmful to the social fabric in an enduring way, nor migration with the same proportion of skilled and unskilled labour as the host country.
Also an immigrant, and I have observed that some immigrants tend to be become more conservative when they move. Or perhaps they have stopped evolving when their peers in the home country have moved on. I think this might be partly due to dealing with the trauma of displacement, and partly due to lack of diversity in thinking. I have mainly noticed this in a generation older than mine.
Europe's struggles seem to be caused more by a culture clash. Immigrants will always bring their way of life with them, and group together - it's a constant of human nature, this is how we preserve identity. Moving to a different country is a traumatic experience when done voluntarily; imagine having to permanently leave home under duress and on short notice.
As an immigrant myself, what always struck me as paradoxical is the nationalist / cultural pride that a lot of immigrants tend to display. I'd sum up my knee-jerk reaction to it as "Bro, if X is so great, why did you leave?" -- again, this is spoken as a person who left their homeland, with first-hand experience of some of the complex issues that surround all that. I still don't quite understand what drives that "flag-waving", but I hope to one day.
Having said all that, I would encourage my adoptive country to be proactive in their immigration policies. Putting it harshly, "get the good ones before they go somewhere else". The geopolitics of the world will only become more chaotic as the impacts of climate change cause hardship in the less habitable areas of the planet -- big migrations are inevitable. If you want your country to arrive in the 22nd century while preserving most of your way of life, you can't be clinging to outmoded ideas.