In general, this isn't the case. There's a certain amount of turnover in the ranks of the very rich because it's so hard to instill discipline in your descendants when they don't have to worry about anything.

Joe Smith works his ass off and gets rich. Maybe his kids are disciplined and smart with their money, but his grandkids are very likely to be popped-collar trustafarians who will piss away the family fortune while not earning shit. Not only that, there's dilution that happens. If you have three kids, your fortune will be split three ways when you die. Not a big deal if you're worth $30 million - lop off half for the estate tax, and $5 million will still set each of your kids for life if they're smart. But what about your grandkids, who are a) undisciplined with money because they're still acting like the family fortune is $30 million, not $5 million and b) only looking at $1 million in inheritance, which is a nice sum of money but isn't enough to buy a yacht?

Rich families become upper-middle class all the time, and they are replaced with families from the upper-middle class.

The poor are, of course, excluded entirely.