Edgar,
I really agree this is a very important health problem. So, what's a feasible study design? The longitudinal follow-up in multiple countries is going to be tough. Perhaps there are ongoing administrative data sets in countries of origin that could be used? Change sin deportation policies (see the other comments on evaluating DACA or ICE activities). It's really important (but not easy) to develop approaches to field rigorous studies that could be done by people without a ton of money!
Maria