Climate change is predicted to alter global species diversity(1), the distribution of human pathogens' and ecosystem services(3). Forecasting these changes and designing adequate management of future ecosystem services will require predictive models encompassing the most fundamental biotic responses. However, most present models omit important processes such as evolution and competition(4,5). Here we develop a spatially explicit eco-evolutionary model of multi-species responses to climate change. We demonstrate that both dispersal and evolution differentially mediate extinction risks and biodiversity alterations through time and across climate gradients. Together, high genetic variance and low dispersal best minimized extinction risks. Surprisingly, high dispersal did not reduce extinctions, because the shifting ranges of some species hastened the decline of others. Evolutionary responses dominated during the later stages of climatic changes and in hot regions. No extinctions occurred without competition, which highlights the importance of including species interactions in global biodiversity models. Most notably, climate change created extinction and evolutionary debts, with changes in species richness and traits occuring long after climate stabilization. Therefore, even if we halt anthropogenic climate change today, transient eco-evolutionary dynamics would ensure centuries of additional alterations in global biodiversity.