This is a problem, and actually precisely that example -- doctors -- was what I was thinking of when I wrote "except for the problem of skills drain". The Economist recently pointed out that this is one way in which the educational systems of the Third World massively subsidize healthcare in the First World.
I'm not sure what the solution is. One idea: perhaps the First World should own up to what they're doing here. They could set aside money to explicitly train doctors, improve medical teaching, etc., in the Third World, and subsidize students with the deal being that they have to agree to remain and work in their country of origin for X years before they can immigrate! (Kind of the way you can get a college education paid for in the USA by agreeing to join the army).