There is a new and excellent paper by Anna Stansbury and Kyra Rodriguez on this topic:

Unlike gender or race, class is rarely a focus of research or DEI efforts in elite US occupations. Should it be? In this paper, we document a large class gap in career progression in one labor market: US tenure-track academia. Using parental education to proxy for socioeconomic background, we compare career outcomes of people who got their PhDs in the same institution and field (excluding those with PhD parents). First-generation college graduates are 13% less likely to end up tenured at an R1, and are on average tenured at institutions ranked 9% lower, than their PhD classmates with a parent with a (non-PhD) graduate degree. We explore three sets of mechanisms: (1) research productivity, (2) networks, and (3) preferences. Research  productivity can explain less than a third of the class gap, and preferences explain almost none. Our analyses of coauthor characteristics suggest networks likely play a role. Finally, examining PhDs who work in industry we find a class gap in pay and in managerial responsibilities which widens over the career. This means a class gap in career progression exists in other US occupations beyond academia.

Here is a first-rate tweet storm by Stansbury on the paper.  Via Aidan Finley.