I think you are concatenating two issues here, as is clear from you referring to "destitute countries" and then using Japan as a prime example!
The flight of skilled professionals from poorer countries to places where they can have a more comfortable life is not just a European problem. There has been a lot of concern recently about many African countries losing such people.
The demographic time bomb in places like Japan is mainly due to the low birth rate, because people who are economically secure don't need to have large families, and maybe also because of the long-hours working culture that leaves people too knackered to reproduce. The latter definitely seems to be the problem in Singapore, which has a very low birth rate; but there the Government is actively encouraging immigration, of all types: skilled professionals, who get to live a very comfortable expatriate lifestyle, and unskilled workers from many other Asian countries, who get poverty wages by Singapore standards but probably more than they would be getting in their home countries. The Singapore population has increased from 3.4m to 5.5m in the last 25 years, despite a fertility rate of 1.2; I'm not sure that people in this country would be happy about immigration on that scale!