Do people migrate to states with more generous Medicaid?
Ok, there is one paper that suggests, yes maybe a little tiny bit. From the abstract:
Results indicate that consumer voters were attracted to states with higher per pupil public school spending, lower property and income tax rates, and that certain consumer-voters may be attracted to states that offer higher levels of Medicaid benefits.
This paper studied pre-ACA data and found a small marginally significant correlation between per-capita medicaid expenditure and net migration. But the paper doesn't use a particularly convincing strategy for causal identification—correlation is not causation, as they say, and in any case the correlation with medicaid was among the weakest they found in the data.
Two other papers have studied the effects of various expansions of medicaid on migration, using much stronger methods for causal inference. Here's Schwartz and Sommers (2014) studying pre-ACA expansions in several states:
Using difference-in-differences analysis of migration in expansion and control states, we found no evidence of significant migration effects. Our preferred estimate was precise enough to rule out net migration effects of larger than 1,600 people per year in an expansion state. These results suggest that migration will not be a common way for people to obtain Medicaid coverage under the current expansion and that interstate migration is not likely to be a significant source of costs for states choosing to expand their programs.
And here's Goodman (2016) with an analysis of the ACA expansion specifically:
Using an empirical model in the spirit of a difference-in-differences, this study finds that migration from non-expansion states to expansion states did not increase in 2014 relative to migration in the reverse direction. The estimates are sufficiently precise to rule out a migration effect that would meaningfully affect the number of enrollees in expansion states, which suggests that Medicaid expansion decisions do not impose a meaningful fiscal externality on other states.
Based on my brief search of the literature these are the only papers on the effect of medicaid on migration specifically, though a much larger literature exists on the effects of welfare programs more generally, usually finding small effect sizes.
The migration literature, including the first paper above, does find that tax policies have a significant effect on migration, with higher taxes causing lower net migration to those localities, consistent with the theory that people move to avoid taxes. It's not hard to guess why taxes would have much larger effects than welfare generosity: the rich who pay taxes can afford to move, while the poor who receive welfare cannot. There are two ways of interpreting this disparity in migration effects. On the one hand, the fact that medicaid doesn't induce migration relaxes incentive compatibility constraints, allowing more efficient redistribution, while on the other hand the high effects of taxation on migration exacerbates these constraints, prohibiting states from being able to finance as much redistribution as they otherwise prefer.