> What should happen is that the existing cultures should be open and welcoming enough so that new-comers want to take part
I'm an immigrant to the EU/NL (naturalized, now a citizen), and I also lived in Sweden for a short stint. If Swedes are anything, they're open and welcoming, I'd even say too open and welcoming, which is part of why these issues keep cropping up.
Some context so I'm not immediately written off as ignorant or as carrying some kind of agenda: I grew up in Indonesia. My family is largely atheist, but we lived in Aceh (look up Qanun Aceh, it's Sharia law territory) and various other places around the country. I've been immersed in Islam, every sect of Christianity, Buddhism (I went to a Buddhist school as a kid), and Balinese Hinduism. I went to an international school with people from literally everywhere, and I'm still close with many of them, scattered across the globe as they now are. I say this to establish that I've actually lived among a wide range of people, cultures, and religions, this isn't an abstract opinion.
This isn't unique to EU immigrant communities either, I've seen the exact same pattern in Indonesia. Many (I want to stress: many, not most, but certainly too many) immigrant groups are remarkably insular and resist integrating in any meaningful way, even multiple generations in. They don't learn the language, don't adopt the host country's customs, and make essentially zero effort to become part of the society they live in. This gets worse where there's already a large existing community of the same background, those often calcify into enclaves where you're functionally not living in the host country at all anymore.
At a certain point, we also have to admit that some immigrants are the source of the problem.
Where I land is this: host countries should expect integration from long-term and permanent residents, and should actively support it via language access, civic education, real investment in helping people become part of the society they're settling into, rather than leaving newcomers to sort it out alone or quietly tolerating permanent parallel communities. What that integration looks like has to be specific to the country in question, because "host culture" isn't one universal template, integrating into Aceh's legal and religious framework is a very different thing from integrating into Sweden's secular, liberal one, and I'm not pretending otherwise.
But in both cases, multiculturalism shouldn't mean a one-way street where only the migrant's culture is expected to remain untouched and is treated as a holy thing while the host culture is expected to bend indefinitely until it, too, warps into something it never was. If you've chosen to build a life somewhere, the society you've moved to should welcome you in, but you should also be expected to actually become part of it. We shouldn't have to walk around on eggshells when we say that people should integrate, this should be the expectation for anyone looking to form a permanent life in a country they live in.
Refugees are a different case, since they didn't choose displacement, but even there once return is no longer realistic and a person has built a long-term life in the host country, the same logic on integration eventually applies to them as well even if the urgency and framing should differ from voluntary migration.
On a side note (and this is more of my own personal problem), I'd rather not see any kind of Sharia-like culture anywhere near me ever again, that crap is stone-age level bullshit that deserves to go extinct forever.