One of the big difficulties with Economics is that one cannot perform controlled experiments like in the physical sciences. While experiments exist, they all have major drawbacks: they may be of too small a scale (although this does not deter medical research to have ridiculous samples), the stakes in the experiment may not be real-life-relevant, controls are to come by, or results may not be easy to generalize. And the larger the problem, the more difficult it is. Think macro-policy, for example.
So I am surprised to find a paper that reviews the literature on migration experiments. You cannot randomize people on whether they migrate or not, or where to. But, as David McKenzie and Dean Yang point out, visa lotteries are exactly that (and decisions of visa officers seem just as random to me...). But only few of those lotteries have been studied, Natural experiments are more common, but about migration per se, rather its impact. For example, the Asian crisis led sharp and unexpected changes in the value of migrant remittances, which impacted families at home. But, of course, one can have questions about how random the "assignment" is across migrants.
As to field experiments, again it is about the impact of remittances. McKenzie and Yang cite as an example a randomized trial where some migrants had access to a savings account in their home country in their name only. This experiment corroborated that migrants have a much stronger savings motive than their families back home. The survey also mentions a paper I discussed before about seasonal famines and migration in Bangladesh.
These are not the experiments I had first in mind, but an interesting introduction nonetheless.