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Interesting analysis, but I think you are discounting the number of people in each of these groups. Percentage basis, yes, the rich and poor will have more children. But both groups are not the 80% of the middle (using your 10% cutoffs on either end).

So I don't think children from the non-rich and non-poor society are going to be aliens in the world anytime soon. Sadly though, fewer will know life with siblings. To meet groups of siblings, you'll find more rich and poor - but only at their birth. Any number of those will change positions in society in any direction.