I'm not really okay with either, but right now most countries do both - what is the point of the multinational if we have to act through the national as well?

You're right, but your example is a neighbouring country - there is a tangible benefit here, that really doesn't exist for most 'donor' nations.

I think that is pretty close, though, with the example of Turkey for the EU. One more step, though: at that point (especially with Turkey/EU,) aren't you just bribing a foreign government to keep refugees there? Like, this is a payment you can never stop - it is paying them tribute in order to convince them to meet their international obligations, isn't it?