Aim Climate change causes shifts in species distributions, or ‘migrations’. Despite the centrality of species distributions to biodiversity conservation, the demonstrated large migration of tropical plant species in response to climate change in the past, and the expected sensitivity of species distributions to modern climate change, no study has tested for modern species migrations in tropical plants. Here we conduct a first test of the hypothesis that increasing temperatures are causing tropical trees to migrate to cooler areas. Location Tropical Andes biodiversity hotspot, south-eastern Peru, South America. Methods We use data from repeated (2003/04–2007/08) censuses of 14 1-ha forest inventory plots spanning an elevational gradient from 950 to 3400 m in Manu National Park in south-eastern Peru, to characterize changes in the elevational distributions of 38 Andean tree genera. We also analyse changes in the genus-level composition of the inventory plots through time. Results We show that most tropical Andean tree genera shifted their mean distributions upslope over the study period and that the mean rate of migration is approximately 2.5–3.5 vertical metres upslope per year. Consistent with upward migrations we also find increasing abundances of tree genera previously distributed at lower elevations in the majority of study plots. Main conclusions These findings are in accord with the a priori hypothesis of upward shifts in species ranges due to elevated temperatures, and are potentially the first documented evidence of present-day climate-driven migrations in a tropical plant community. The observed mean rate of change is less than predicted from the temperature increases for the region, possibly due to the influence of changes in moisture or non-climatic factors such as substrate, species interactions, lags in tree community response and/or dispersal limitations. Whatever the cause(s), continued slower-than-expected migration of tropical Andean trees would indicate a limited ability to respond to increased temperatures, which may lead to increased extinction risks with further climate change.