When did the change happen? For instance, my sixth great grandparents were farmers in the mid west (Missouri). They had 12 children of their own. So, for my family, the poor having fewer children than the rich probably would have happened farther back than the late-1700s/early-1800s.

For England I want to say the crossover is about 150 years ago, could be 200, from memory. France & Germany a generation or two later, most other places later still.

The US frontier may well be different -- having almost infinite land meant it could sustain rapid expansion for a long time. I know that New England & Quebec had something like a doubling per generation in total a bit earlier, from say 1650-1800. Which was unprecedented, I don't think any other pre-industrial civilisation got close. (Mathus had the numbers and was suitably impressed.)

(Bear in mind that number of siblings looking backwards is a different measure to number of offspring looking forwards -- 2 kids/parent on average is perfectly compatible with everyone having 3 siblings, as long as they have some childless aunts.)

https://en.wikipedia.org/wiki/British_Agricultural_Revolutio... is the primary cause of this change. Until food productivity started growing dramatically, populations were generally kept in check by the threat of famine. And famine was more a problem for peasants than nobles.

The change happened earlier in the Americas for the simple reason that due to recent settlement, population density did not yet match what the land could sustain.

Certainly these improvements led to overall population growth, which accelerated over a few centuries. But the pattern that richer people had more kids than poor is a different thing. I believe this persisted until Victorian times in England. The US frontier was a gigantic anomaly in other ways, and could be in this way too, I don't think I've seen data.