None of this actually requires immigration. Cultural imports are enough to utterly destroy aspects of local culture without a single person immigrating.

I don't know about Norway or Iceland specifically, but I have been told that English is so trendy in Sweden that not a lot of cultural works are actually written in Swedish anymore. As for China, we've actually gotten cultural backwash from Hollywood doing so much business there. Specifically, a lot of western media companies have had to self-censor in order to retain Chinese market access.

If we really wanted to coat every culture in amber then we would need a lot more than just immigration controls. We would need to emulate pre-Meiji Japan in every country in the world: i.e. no migration whatsoever, and heavy restriction and censorship of cultural imports. So no Internet, no imported cinema or TV shows, no international phone calls... hell, no international postal system.

Less dramatically you have countries like Canada or France that have local content quotas, but those have their own issues. But those are less "stop cultural change" and more "keep Hollywood from completely smothering our local creative industry".