Explosive radiations have been considered one of the most intriguing diversification patterns across the Tree of Life, but the subsequent change, movement and extinction of the constituent species makes radiations hard to discern or understand as geological time passes. We synthesised phylogenetic and distributional data for an ongoing radiation – the mega-diverse plant genus Solanum L. – to show how dispersal events and past climatic changes have interacted to shape diversification. We found that despite the vast diversity of Solanum lineages in the Neotropics, lineages in the Old World are diversifying more rapidly. This recent explosive diversification coincides with a long-distance dispersal event from the Neotropics, at the time when, and to places where, major climatic changes took place. Two different groups of Solanum have migrated and established in Australia, but only the arid-adapted lineages experienced significant increases in their diversification, which is consistent with adaptation to the continent9s long-term climatic trend and the diversification of other arid-adapted groups. Our findings provide a clear example of how successful colonisation of new areas and niches can - but do not always - drive explosive radiations.