More importantly, even small changes in demand means immigrants can and will be more picky about where to go.
Growth in their original countries also means a reduced supply of willing candidates. E.g. the UK saw that with Polish immigrants in the building trade who started returning as salaries increased at home, despite the disparity still being huge.
As the demographics shift starts hitting more countries, it's not just that the total number of people in working age in those countries drop, but the proportion of them who has it financially bad enough at home to want to leave drops.