Understanding the processes that have generated the latitudinal biodiversity gradient and the continental differences in tropical biodiversity remains a major goal of evolutionary biology. Here we estimate the timing and direction of range shifts of extant flowering plants (angiosperms) between tropical and non-tropical zones, and into and out of the major tropical regions of the world. We then calculate rates of speciation and extinction taking into account incomplete taxonomic sampling. We use a recently published fossil calibrated phylogeny and apply novel bioinformatic tools to code species into user-defined polygons. We reconstruct biogeographic history using stochastic character mapping to compute relative numbers of range shifts in proportion to the number of available lineages through time. Our results, based on the analysis of c. 22,600 species and c. 20 million geo-referenced occurrence records, show no significant differences between the speciation and extinction of tropical and non-tropical angiosperms. This suggests that at least in plants, the latitudinal biodiversity gradient primarily derives from other factors than differential rates of diversification. In contrast, the outstanding species richness found today in the American tropics (the Neotropics), as compared to tropical Africa and tropical Asia, is associated with significantly higher speciation and extinction rates. This suggests an exceedingly rapid evolutionary turnover, i.e., Neotropical species being formed and replaced by one another at unparalleled rates. In addition, tropical America stands out from other continents by having "pumped out" more species than it received through most of the last 66 million years. These results imply that the Neotropics have acted as an engine for global plant diversity.