Mountain ranges are amongst the most species-rich habitats, with many large and rapid evolutionary radiations. The tempo and mode of diversification in these systems are key unanswered questions in evolutionary biology. Here we study the Andean Lupinus radiation to understand the processes driving very rapid diversification in montane systems. We use genomic and transcriptomic data of multiple species and populations, and apply phylogenomic and demographic analyses to test whether diversification proceeded without interspecific gene flow - as expected if Andean orogeny and geographic isolation were the main drivers of diversification - or if diversification was accompanied by gene flow, in which case other processes were probably involved. We uncover several episodes of gene flow between species, including very recent events likely to have been prompted by changes in habitat connectivity during Pleistocene glacial cycles. Furthermore, we find that gene flow between species was heterogeneously distributed across the genome. We argue that exceptionally fast diversification of Andean Lupinus was partly a result of Late Pleistocene glacial cycles, with associated cycles of expansion and contraction driving geographic isolation or secondary contact of species. Furthermore, heterogeneous gene flow across the genome suggests a role for selection and ecological speciation in rapid diversification in this system.