That's just false! In every society, the elite class is overwhelmingly comprised of families who maintain their high social standing and wealth over decades or centuries. The lower classes are similarly static over the generations, and all classes have a class-specific bias.
If you need proof, take a sampling of the 'Early Life' Wikipedia section of any elite you can think of. Even tech founders, who are more likely socially mobile than most other elites, have overwhelmingly had an upper-middle class upbringing at _least_.