Regional species diversity is ultimately explained by speciation, extinction, and dispersal. Here we estimate dispersal and speciation rates in Neotropical rainforest biomes to propose an explanation for the distribution and diversity of extant butterfly species. We focus on the tribe Brassolini (owl butterflies and allies): a Neotropical group that comprises 17 genera and 108 species, most of them endemic to rainforest biomes. We infer a total-evidence species tree using the multispecies coalescent framework. By applying biogeographical stochastic mapping, we infer ancestral ranges and estimate rates of dispersal and cladogenesis at the scale of millions of years. We suggest that speciation in Mesoamerica and the northwestern flank of the Andes have only increased within the past 2 million years. In contrast, speciation in the Brazilian Atlantic Forest has been constant throughout the past 10 million years. The disparate species diversification dynamics may be partly explained by the geological and environmental history of each bioregion. Importantly, the dispersal rates into the Atlantic Forest and Mesoamerica plus NW Andes increased simultaneously in the middle-Miocene, suggesting that lineages from such regions have had comparable times for speciation despite their decoupled diversification dynamics. Diversification of extant Amazonian lineages, on the other hand, has episodically increased since the late Miocene, including a rise in speciation rate during the Pleistocene. Altogether, our results reveal a mosaic of biome-specific evolutionary histories within the Neotropics, where species have diversified rapidly (cradles: e.g., Mesoamerica), have accumulated gradually (museums: e.g., Atlantic Forest), or have alternately diversified and accumulated (e.g., Amazonia).