Estimating population spread rates across multiple species is vital for projecting biodiversity responses to climate change. A major challenge is to parameterise spread models for many species. We introduce an approach that addresses this challenge, coupling a trait-based analysis with spatial population modelling to project spread rates for 15 000 virtual mammals with life histories that reflect those seen in the real world. Covariances among life history traits are estimated from an extensive terrestrial mammal data set using Bayesian inference. We elucidate the relative roles of different life-history traits in driving modelled spread rates, demonstrating that any one alone will be a poor predictor. We also estimate that around 30% of mammal species have potential spread rates slower than the global mean velocity of climate change. This novel trait-space-demographic modelling approach has broad applicability for tackling many key ecological questions for which we have the models but are hindered by data availability.