"Furthermore, immigrant parents were “under-placed” in the income distribution, thus allowing their children who were native English speakers and educated in the US more scope for upward mobility."
I don't see numbers on how much of the total "upward mobility" can be attributed to any of the individual factors.
The children of upward mobile immigrants who don't come from lower classes in their home country reflect that, even if the parents struggle in the new country.
They are socialized differently from true poor people.
For instance, language: the children will tend to pick up the "socio-economically preferred dialect"¹ of English, rather than a socially stigmatized dialect. The children will do this naturally, because they will tune in to how the way people speak English in America determines the perception of social class, and then they will learn that way of speaking which fits their self-image about where they fit in society.
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1. A phrase I picked up years ago from a linguistics prof.