My experience living in an incredibly multi-cultural immigrant dense area, is overwhelmingly "No". People like their "home" cultures (generally they believe its the best), and they tend to carry some degree of resentment for their new society for not recognizing that. It creates endless friction at the edges (whereas core values tend to be pretty similar globally).
Japan doesn't really deal with this at all, and what immigrant ingress they have is usually fueled by people trying to live a Japanese like life.
Caving on manners to avoid the "struggle", or rather, caving on manners because one benefits from doing so is just a reflection of how much one values the values. This is simply saying that more often than not we can maintain them, it just might not be worth it to us.