If your whole neighborhood goes up, but not the whole city/municipality, correct. If the entire tax base goes up (not just one neighborhood), you should expect the mill rate to fall.
Tax rates are generally changed by an independent political process (the municipal budget) which often has the power to increase/decrease the mill rate based on how much money is needed/desired. This doesn't change the fact that the mill rate does generally fall by a lot when properties are reassessed in areas that have seen a large property price increase.
Yeah I'm very curious about this. I assumed that because my taxes have been purely proportional in the several places that I've lived that this was generalized to across the U.S.
This is the first I've heard of a city adjusting taxes down.