Understanding and predicting the evolutionary responses of complex morphological traits to selection remains a major challenge in evolutionary biology. Because traits are genetically correlated, selection on a particular trait produces both direct effects on the distribution of that trait and indirect effects on other traits in the population. The correlations between traits can strongly impact evolutionary responses to selection and may thus impose constraints on adaptation. Here, we used museum specimens and comparative quantitative genetic approaches to investigate whether the covariation among cranial traits facilitated or constrained the response to selection during the major dietary transitions in one of the world’s most ecologically diverse mammalian families—the phyllostomid bats. We reconstructed the set of net selection gradients that would have acted on each cranial trait during the major transitions to feeding specializations and decomposed the selection responses into their direct and indirect components. We found that for all transitions, most traits capturing craniofacial length evolved toward adaptive directions owing to direct selection. Additionally, we showed instances of dietary transitions in which the complex interaction between the patterns of covariation among traits and the strength and direction of selection either constrained or facilitated evolution. Our work highlights the importance of considering the within-species covariation estimates to quantify evolvability and to disentangle the relative contribution of variational constraints versus selective causes for observed patterns.