The density of species varies widely across the earth. Most broad taxonomic groups have similar spatial diversity patterns, with greatest densities of species in wet, tropical environments. Although evidently correlated with climate, determining the causes of such diversity differences is complicated by myriad factors: many possible mechanisms exist to link climate and diversity, these mechanisms are not mutually exclusive, and they may overlap in the patterns they generate. Further, the importance of different mechanisms may vary between spatial scales. Generating uneven spatial diversity patterns in regions that are below equilibrium species richness requires either geometric or historical area effects, or regional differences in net diversification. Here, I investigate the global climate correlates of diversity in plants and vertebrates, and hypotheses that could link these correlates to net diversification processes, in particular through climate-linked patterns of molecular evolution. I first show strong climate–diversity relationships only emerge at large scales, and that the specific correlates of diversity differ between plants and animals. For plants, the strongest large-scale predictor of species richness is net primary productivity, which reflects the water–energy balance at large scales. For animals, temperature seasonality is the strongest large-scale predictor of diversity. Then, using two clades of New World passerine birds that together comprise 20% of global avian diversity, I investigate whether rates and patterns of molecular evolution can be linked to diversification processes that could cause spatial diversity patterns in birds. I find that most substitution rate variation between phylogenetically independent comparisons of avian sister species appears to result from mutation rate variation that is uncorrelated with climate. I provide evidence of nearly neutral effects in mitochondrial coding sequences, finding a significant, negative correlation between non-synonymous substitution rates and population size. Using phylogenetically independent comparisons, I also find that birds in low temperature seasonality, and isothermal environments, and birds with small elevational ranges have increased non-synonymous substitution rates, indicative of relaxed purifying selection. Other climate variables have no direct effect on molecular evolution. Molecular evolutionary patterns are dominated by mutation rate variation. Recovered patterns were stronger when mutation rate variation was controlled, indicating that such variation is a source of noise in analyses, and may be generally problematic across short genetic distances for analyses using mitochondrial genes. I bring these findings together with emerging literature to outline a framework for understanding net diversification patterns. Maintaining adaptations to climate, and the limits of those adaptations have population-genetic consequences that can affect lineage persistence and the processes of speciation and extinction in a fashion that is consistent with observations at multiple levels of diversity.