which makes a lot of them keep their old house when they move, and just rent it out.

Yeah, that's not a thing:

Meet the Billionaire Investors Behind the US Housing Affordability Crisis

With roughly 800 billionaires in the U.S. with combined wealth of $6.2 trillion (and 2,781 billionaires globally with over $14.2 trillion), ultra-wealthy investors tend to diversify their holdings across multiple kinds of assets. A huge amount of this billionaire wealth is invested in property, land, and housing. Billions and possibly trillions of dollars are sucked into predatory investment practices and luxury housing schemes — where global billionaire investors park vast quantities of wealth in U.S markets.

This is not your grandparent’s gentrification, but rather a hyper-gentrification fueled by concentrated wealth driving up land and housing costs, expanding short-term rentals, and treating housing like a commodity to speculate on or a place to park wealth. The billionaires are displacing the millionaires, and the millionaires are disrupting the housing market for everyone else.