The causes of continental patterns in species richness continue to spur heated discussion. Hypotheses based on ambient energy have dominated the debate, but are increasingly being challenged by hypotheses that model richness as the overlap of species ranges, ultimately controlled by continental range dynamics of individual species. At the heart of this controversy lies the question of whether species richness of individual grid cells is controlled by local factors, or reflects larger-scale spatial patterns in the turnover of species' ranges. Here, we develop a new approach based on assemblage dispersion fields, formed by overlaying the geographic ranges of all species co-occurring in a grid cell. We created dispersion fields for all tetrapods of South America, and characterized the orientation and shape of dispersion fields as a vector field. The resulting maps demonstrate the existence of macro-structures in the turnover of biotic similarity at continental scale that are congruent among vertebrate classes. These structures underline the importance of continental-scale processes for species richness in individual assemblages. Ecologists continue to debate whether local species assemblages result from habitat filtering or from turnover among the regional species pool. Here the authors develop a "dispersion field" method to mapping species range overlaps, showing that regional turnover processes are key to local assembly.