Ecological communities are assembled from the overlapping of species in geographic space, but the mechanisms facilitating or limiting such overlaps are difficult to resolve. Here, we combine phylogenetic, morphological and environmental data to model how multiple processes regulate the origin and maintenance of geographic range overlap across 1,115 pairs of avian sister species globally. We show that coexistence cannot be adequately predicted by either dispersal-assembly (that is, biogeographic) models or niche-assembly models alone. Instead, our results overwhelmingly support an integrated model with different assembly processes dominating at different stages of coexistence. The initial attainment of narrow geographic overlap is dictated by intrinsic dispersal ability and the time available for dispersal, whereas wider coexistence is largely dependent on niche availability, increasing with ecosystem productivity and divergence in niche-related traits, and apparently declining as communities become saturated with species. Furthermore, although coexistence of any individual pair of species is highly stochastic, we find that integrating assembly processes allows broad variation in the incidence and extent of coexistence to be predicted with reasonable accuracy. Our findings demonstrate how phylogenetic data coupled with environmental factors and functional traits can begin to clarify the multi-layered processes shaping the distribution of biodiversity at large spatial scales.