The two main extensions of rain forest in South America are the Amazon and the Atlantic rain forest (Mata Atlantica) of Brazil. These are separated by a wide ‘dry diagonal’ of seasonal vegetation formations, which includes the dry forests of the Brazilian caatinga and the savannas of the cerrado. Here, we used Inga, a species-rich genus of trees in neotropical rain forests, to study past connections between the Amazon and Mata Atlantica, in order to test the idea that dispersals between the two forests have been clustered during specific time periods corresponding to past, humid climates. We performed hybrid capture sequencing of 810 nuclear loci for 453 Inga accessions representing 164 species that included 62% of Mata Atlantica species and estimated dated phylogenies using maximum likelihood for all accessions and a species-level tree using coalescent methods, with subsequent temporal calibration. We find that there have been 18-21 colonisations of the Mata Atlantica from the Amazon, with only one dispersal in the reverse direction. These dispersal events have occurred over the evolutionary history of Inga, and we find no evidence for clustering of dispersal events into any specific time period, both based on comparing models of alternative biogeographic histories and null simulations that show the temporal distribution of dispersal events matches a random expectation. Time-specific biogeographic corridors do not need to be invoked to explain dispersal between the Amazon and Mata Atlantica for rain forest trees such as Inga, which are likely to have used a dendritic net of gallery forests to cross the dry diagonal.