Especially lower child mortality rates. When people are reasonably confident of seeing all of their children grow to adulthood, they just have fewer of them.

I'm sorry but this is just incorrect. Education, specifically the education rates of women, is the leading indicator... all other correlations are merely side effects of this.

I would love to see some research touching upon the finer points of this, I've always assumed child mortality was the leading factor. The education thing I think might be some causal reversal, women have time to study because they have less children to take care of. She can study (as well as invest in daycare) because she is more wealthy? This is just armchair theorizing, haven't bothered to look it up.