Progress on the model. I've now managed to prove that if the democracy makes it more costly to immigrate... then fewer people migrate. Never let it be said that formal theory can't generate powerful counter-intuitive predictions.
(Why isn't this trivial? Well, substantively it is trivial of course. Formally, the problem is that migration is partly a collective decision. If you think lots of skilled workers are going to migrate and increase the democratic tax base, then you as an unskilled worker might want to migrate with them. Skilled workers face slightly more ambiguous incentives: if many of them are migrating, the tax base will increase but that might also give the unskilled median voter an incentive to put taxes up. So, before I can say anything about how migration costs affect the game, I have to work out the equilibrium.)
The next step is showing more interesting things. The essential idea is: if there are a lot of unskilled workers in the dictatorship, the democratic median voter will put immigration costs up to deter them from coming (while allowing some skilled workers to come). Then the dictator will take advantage of his subjects' reduced incentives to leave by increasing taxes.
I lost a day this week. I woke up today and wondered why my alarm hadn't gone off. Then my phone told me it was Saturday. Where the hell did Friday go? Apparently it was so like Thursday (and the day before that...) that I didn't even notice it. Give me my day back, dammit!