Poking around looking for stuff about immigration and welfare, I found this 1997 Dallas Fed paper by Madeline Zavodny. Here's her conclusion:
Much of the motivation for eliminating most immigrants’ access to federally funded public assistance benefits was concern that persons migrate to the United States because of the availability of welfare benefits. The 1996 welfare law makes noncitizens ineligible for food stamps and SSI payments and allows states to discontinue AFDC, Medicaid, and other public assistance benefits to noncitizens. Several states intend to continue extending benefits to noncitizens, whereas others are likely to cut off benefits, widening the already substantial differences in welfare benefits across states. These differences in policy create concern that immigrants will move in response to interstate differentials and that states that continue to allow immigrants to receive welfare payments will become welfare magnets.
In this article, I find little evidence to support the contention that new immigrants will choose their destinations based on welfare generosity. New immigrants are attracted to areas with large immigrant populations. Because earlier immigrants are disproportionately located in high-welfare states, it may appear that high welfare benefits attract immigrants. However, immigrants do not respond to interstate differentials in welfare generosity but rather to differences in the sizes of the foreign born populations. Immigrants are also attracted to a specific subset of states—namely California, New York, Florida, and Texas—and do not respond to changes in welfare benefits within states over time. The recent historical evidence gives little reason to be concerned that new immigrants will choose their destinations based on the welfare differentials created by the new welfare law.
This is notable for two reasons. First, it seems like most people don't know that the 1996 welfare reform made most noncitizens ineligible for most federal benefits. I didn't either, until fairly recently. Second, it supports the idea that welfare isn't a significant factor in immigrants' choices about where to go. Maybe this has changed in the last 10 years; I'd like to see a more recent study. But I'd guess this holds up.