500yr is waaaaay too long. 100yr ago nobody in my family tree spoke English and they were all subsistence farming on a different continent. Every relative I know is decently prosperous. Some people make more of their lives than others but nobody is disadvantaged.

I have a friend who's grandparents had everything taken by the Japanese, then again by the communists, moved somewhere they couldn't even speak the language (they got called crazy but they got the last laugh when everyone else starved) and my friend owns a house in a gentrifying city and makes six figures sitting on his butt staring at a screen. Not bad for two generations.

I knew a woman who lost a good chunk of her immediate family to violence in a south American country her mom moved with her to the US (I'm pretty sure she came to the US as a refugee) they eventually wound up in one of the "worse" cities in NJ and her mom signed her up for some educational program that somehow led to her being sent to a prep school in New England and from there she wend to college and graduated from Colombia.

Compare all of those to my girlfriend's family tree which is chock full of deadbeats. I don't know a single one who is actively engaged in working hard to move up in the world. They've been on this continent longer than the existence of the nation they reside in. Maybe they made something of themselves once upon a time but this branch of the family tree has done nothing productive.

I think whether or not you grow up in a household with parents who are driven to raise their kids well is the primary determining factor. If colleges want to know how "disadvantaged" a kid is they should be looking at the parents. Immigrants for the most part tend to be very industrious and pass that on to their children and grandchildren but it seems to diminish over the generations and the variance among individuals takes over as the determining factor. Some individuals are highly driven. Some are deadbeats.