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Misha comments on Decision Theory Paradox: PD with Three Implies Chaos? - Less Wrong
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They would not have far more children in the long run. Their descendants would have more children, but their utility function doesn't care about that.
Edit: and if they do start caring about grandchildren the problem stops being a straightforward Prisoners' Dilemma.
So can we show that TDT would play perfectly in such a scenario? I think yes.
Your decisions interact in complex ways with your peer's decisions, your descendant's decisions, etc. There might be a nice simple formula that classifies this, but:
These are TDT agents. So they all behave the same. At least, the ones during the same timestep behave the same. The ones during different time steps are in different situations, but have the same values. Since they (overall) have he same values as past generations, they will together implement a coherent strategy.