The latitudinal diversity gradient (LDG), where the number of species increases from the poles to the Equator, ranks among the broadest and most notable biodiversity patterns on Earth. The pattern of species-rich tropics relative to species-poor temperate areas has been recognized for well over a century, but the generative mechanisms are still debated vigorously. We use simulations to test whether spatio-temporal climatic changes could generate large-scale patterns of biodiversity as a function of only three biological processes—speciation, extinction and dispersal—omitting adaptive niche evolution, diversity-dependence and coexistence limits. In our simulations, speciation resulted from range disjunctions, whereas extinction occurred when no suitable sites were accessible to species. Simulations generated clear LDGs that closely match empirical LDGs for three major vertebrate groups. Higher tropical diversity primarily resulted from higher low-latitude speciation, driven by spatio-temporal variation in precipitation rather than in temperature. This suggests that spatio-temporal changes in low-latitude precipitation prompted geographical range disjunctions over Earth’s history, leading to high rates of allopatric speciation that contributed to LDGs. Overall, we show that major global biodiversity patterns can derive from interactions of species’ niches (fixed a priori in our simulations) with dynamic climate across complex, existing landscapes, without invoking biotic interactions or niche-related adaptations.