Migration: so I simplified the model a bit, on a suggestion from Guy, and it now looks more tractable. The aim is just to show that the larger the initial difference between a democracy and a dictatorship, in terms of citizens' productivity, the higher the barriers set to migration by the democracy's median voter, and the higher the resulting tax/theft rate of the dictator. I was spending too much time fiddling with the question of whether the dictator might not be better off setting up a Singapore-style tax haven for rich democrats. Sometimes that will happen, but it's not a central concern.
So I'm just going to assume that migration is one way; and I'm going to have only two types of workers (high and low productivity), with different mixes in each country. Then the democracy's median voter always has an incentive to bring in high productivity workers - up until they are just a bare minority, so that the unskilled still control the political outcome. Having only two types makes it easier to parametrize differences in productivity between countries, and that's what I need to get some comparative statics going.
Cheap talk/ideology: still no traction. I need a way for message meanings to be fixed in some sense, and that is hard in a model where players are rational.
Life: I am utterly, utterly skint. I can't even buy myself a coffee at the moment. God knows how I'll survive until hitting Miami (w00t!) at the end of the month.