That's a good solution for english/latin speaking countries since there's a large pool of migrants to choose from which already know the language but for a country like Japan with its niche language and specific customs.. it's much tougher unless they wish to change their culture which they probably don't and having migrants assimilate into their culture isn't an easy ask either.

It's also only a viable solution for countries with small/medium populations. As India and China start ageing it's going to be hard for them to find the hundreds of millions of migrants required.

I'd imagine China will indeed start to pull out all stops to attract younger migrants at some point, and I'd guess many may even come from India, and certainly from large countries with high fertility like Nigeria. But sure, it can only help so much if too much of the world's population is living in countries with unbalanced demographics.

By that time India will get close to par to China in per capita GDP, and will hopefully still remain democratic. Why would Indians do that?

Even if India's per capita GDP has largely caught up, there'd still be a substantial underclass of young workers who are likely to be attracted by higher incomes they could get in China, given the differences in income distribution. And migrants nearly always sacrifice a certain amount of democratic franchise just by virtue of not living in a country where they have citizenship - it doesn't seem to be a great deterrent, compared to other cultural/language barriers and having to live far from family etc.

All of those worker source countries are facing their own aging problems. It’s not a solution.

They may eventually. And no, it's not a long term global solution, but it buys enough time to avoid the worst impacts in a real demographic crunch. You are right though that, for instance, the outflux of healthcare works from countries like the Phillipines to Australia and other nations with greater income potential does cause strain on their own healthcare systems.