Classical niche-assembly models propose that the composition of biotic communities in continental landscapes is determined chiefly by the autecology of species, interspecific competition, and the diversity of resources and habitats within a region. In contrast, stochastic models propose that simulation algorithms can replicate the macroecological patterns, if not the mechanisms, of community assembly. Despite fundamental differences in assumptions, both categories of models assume that species are drawn from regional source pools. We explored the implications of source pool geometry on the assembly of avian communities with an analysis of assemblage dispersion fields, which can be visualized by overlaying the geographic ranges of all species that occur in an assemblage. Contours of species richness surrounding focal quadrats illustrate the decay rate of assemblage similarity with distance and the probable geometry of assemblage source pools. We used a geographic database for 2,891 species of South American birds to characterize dispersion fields for assemblages sampled by 1 degrees latitude-longitude quadrats (n = 1,676). We show that the median range size of dispersion fields varies by an order of magnitude across the continent. Because abundance generally correlates with geographic range size within taxonomic groups, the number of individuals per species in avifaunal source pools must also vary by an order of magnitude. Most significantly, dispersion field geometry was surprisingly asymmetrical and exhibited complex geographical patterns that were associated with the distribution of biomes. These results are broadly consistent with the predictions of niche-assembly models but offer little support for stochastic assembly models.