Note that richer countries have fewer children per capita than poor ones, and richer demographics within those countries tend to fewer children than poorer ones.

I can think of several reasons offhand why rich people would have fewer children:

1. The lifestyle conducive to being rich makes the parents more likely to delay childbearing, possibly leading to having fewer children.
2. Rich people are smarter, more educated, and/or have better impulse control and therefore are more likely to use family planning properly.
3. Raising a child is disproportionately costly for richer people--they may spend lots of money on the child for things like college educations, and they don't have jobs such as farming which would let them benefit from using the kids as manual labor.

The first factor applies to the degree that becoming rich is controlled by the parents' actions. The second applies to the degree that being rich is associated with the parent's traits. And the third applies to the degree that the parents want to achieve a particular level of wealth for their children and need to spend money to do so. Having the children become marginally richer because of factors that are not related to the parents' actions or traits and do not involve spending money on them would not lead to that marginal increase being correlated with having fewer children.