Many biological phenomena can be thought of, and modeled as generalized population structure, with individuals belonging to different states. Certain groups of states in such models are often connected by ``migration'' that occurs on a time scale that is much faster than the coalescent time scale. We show that, when viewed on the coalescent time scale, the structure associated with such groups of states collapses, and is replaced by a kind of averaging over the states. When the entire structure collapses, the standard coalescent is retrieved. The effect of the population structure on the coalescent is then captured by a simple scaling factor, that is related to various notions of ``effective population size.'' It is also possible for parts of the structure to collapse, leading to a simpler structured coalescent.