Dynamics, like income mobility dynamics are a complicated thing. We tend to think in stories, and that is useful, but it makes it easy to reach overly general conclusions.

The old republics (US & France, mostly) tend to overlook class... to some extent intentionally. My country (ireland) has the opposite problem: too much class consciousness. To much telling of class-sentric stories.

But, class dynamics do exist in the US & France too.

Class mobility exists too, but it's nowhere near as mobile as income mobility. Class is stickier by definition. You can just as easily colour in a class stickiness story with statistics. The likelihood at birth of someone graduating high school, going to prison, being poor, rich etc. It is true that where you start is a pretty good predictor, statistically, of where you will end up.

It's important to keep both perspectives in mind, simultaneously. There are real opportunities to escape poverty. The ovarian lottery is also very important, so are other "lotteries." It's also a mistake to see yourself as a statistic.