Most climate change predictions omit species interactions and interspecific variation in dispersal. Here, we develop a model of multiple competing species along a warming climatic gradient that includes temperature-dependent competition, differences in niche breadth and interspecific differences in dispersal ability. Competition and dispersal differences decreased diversity and produced so-called 'no-analogue' communities, defined as a novel combination of species that does not currently co-occur. Climate change altered community richness the most when species had narrow niches, when mean community-wide dispersal rates were low and when species differed in dispersal abilities. With high interspecific dispersal variance, the best dispersers tracked climate change, out-competed slower dispersers and caused their extinction. Overall, competition slowed the advance of colonists into newly suitable habitats, creating lags in climate tracking. We predict that climate change will most threaten communities of species that have narrow niches (e.g. tropics), vary in dispersal (most communities) and compete strongly. Current forecasts probably underestimate climate change impacts on biodiversity by neglecting competition and dispersal differences.