The population genealogical processes associated with a wide range of exchangeable reproductive models (including the Wright-Fisher model) are shown to converge weakly, as the population size becomes large, to a particularly tractable limiting process, the age-ordered analog of Kingman's coalescent. This result extends the known convergence results for sample processes and effectively completes the robustness theory for neutral genealogies. Its consequences, which include a unification of the results for neutral models, have already been exploited elsewhere. The techniques used rely heavily on knowledge of sample behavior, together with consistency arguments. They may be of more general interest.