Unfortunately this answer is not simple... Or I should say, it's only a short term solution.

What happens when the places that are making lots of people now, stop?

Us: "Hey India, send me some more people"

India: "Sorry dude, we're out"

India currently is at 2.05 fertility rate, which is replacement rate. Now for quite some time they will have excess population in relation to their wealth, but I don't expect that to last forever.

If there aren't enough young people to take care of the old, this is directly the fault of the prior generations. Importing labor to take care of them (indirectly) would be a ridiculous overvaluation of their interests over the younger generations' interests. Permanent change to solve a fundamentally temporary problem.

Most immigrant groups don't generate engineers and scientists at the same rate that western countries.