I think if you change the time period and place and replace "Asian" with whatever cultural group makes up the majority of immigrants in that situation, you get a similar outcome:
1. First generation deals with incredibly strong prejudice, sometimes clear and overt but sometimes disguised as "they're not trying hard enough to assimilate" or "they're stealing our jobs". But they've come from a situation of poverty or extremely limited opportunity, so they work like crazy and push their children to take advantage of the new country's opportunity (in practice, this usually means education). In Canada right now, this would be mostly folks from the Middle East and sub-Saharan Africa and China.
2. Second generation often feels pressured when they compare themselves to native-born peers, but often kick ass in school and build interesting hybrid friendship groups and identities. They face prejudice, but it's lowered because they have no "foreign" accent and they understand the new country's social norms. This is my generation, and in Canada right now I guess would be mostly people with ethnic roots in the subcontinent or Caribbean, descendants of the Vietnamese boat people, etc.
3. Third generations are fairly thorough assimilated to the new culture. They hang on tight to (often mostly symbolic) aspects of their grandparents' culture (food, festivals, folkdance!), but often don't speak the language. By this point the host society has pretty well gotten used to them, so most remaining prejudice comes from people most of us would uncontroversially describe as "racist assholes". In Canada: Italians, Greeks, Poles, early-arriving brown folks (because Canada's immigration policies became much less racist around 1960).
4. Fourth and subsequent generations are assimilated. In Canada, this is made easier by the fact that most of those folks are phenotypically "white", but even that description comes in part from changing ideas of who's inside whiteness. See the book "How the Irish Became White". In Canada: eastern European Jews, Germans, Irish, Ukrainians. At this point, people end up distributed across the spectrum wrt success, based almost entirely on skills, ability, effort, etc... as a Black friend of mine says, "if you're white and your people have been here for six generations, and you're not rich, somebody's fucked up."
I realize I'm replacing ethnic stereotyping with generational and historical stereotyping or simplification here, but I still think something like this holds pretty well. The process seems remarkably clear in Australia, where the country's need for (and unfamiliarity with) immigrants in past generations has led to really overt government efforts to introduce old-school to new "exotic" foreigners, which started with... the Finns!