Could there be any plausible reasons why citizens of a rich developed country would want to emigrate?
I need reasons strong enough that country would constantly suffer from large emigration.
There are several, very different, possible reasons; I will just sketch a few, bear in mind several variations are possible for each one.
Money isn't everything, you know. Suppose the non-rich country offers a better quality of life than the rich one? Basically the same reason I choose to live in a non-rich state (and in a rural area) despite the fact that I would have made a lot more money if I'd stayed in Silicon Valley.
Beyond that, with telecommuting you don't have to live where you work. I've worked in the US while living in Europe, and vice versa. So if you can telecommute, you can live in a nice but job-free place, while working in a nasty but high-paying one.
A final reason, and one that a number of people actually do, is that retiring in a place with a low cost of living lets your money go futher. A pension that is barely sufficient in the US or Britian goes a lot further in say Costa Rica or Spain.