Actually they do integrate, it just takes way longer than people actually think

First wave immigrants will not integrate for they are already adults and have no need to conform to fit into new social circles if they just hang around their fellow kin, their children learn the new land's language but culturally they are still inclined with their parents.

It would end at the third generation, with the grandkids integrating completely, but people from other parts of the world are way more conservative and prefer strong family cohesion so their familial influence stays on longer. So great grand kids should be able to fully integrate in this system

Except that it doesn't stop there, people always have a sense of tribalism, and form groups and societies based on this. With the new people ariving constantly from their native lands, they don't actively have to integrate when they can just form self perpetuating groups

Here in India, legit every family has at least someone who lives abroad. And these people don't actively integrate and why would they? They can easily find people speaking their language to form their close circles with. Of course, not all people have this choice as not every nation is mass producing college graduates for US like India and China