Great comments, glad you liked the papers! Can you explain more about what you mean by your question about if health selection always increases with worsening health conditions? Do you mean in the sending country?
I agree that the finding that the effect does not last for a long time or into the next generation is interesting. There is also interesting research in the fertility literature on selection by fertility status and how fertility changes with time/generations among immigrants. See this paper on immigrants from mexico to the US https://www.jstor.org/stable/pdf/26348215.pdf
And then slightly different patterns in Europe:
https://www.jstor.org/stable/pdf/26727035.pdf
I agree that the finding that the effect does not last for a long time or into the next generation is interesting. There is also interesting research in the fertility literature on selection by fertility status and how fertility changes with time/generations among immigrants. See this paper on immigrants from mexico to the US https://www.jstor.org/stable/pdf/26348215.pdf
And then slightly different patterns in Europe:
https://www.jstor.org/stable/pdf/26727035.pdf