Adaptive radiations are hypothesised as a generating mechanism for much of the morphological diversity of extant species^{1,2,3,4,5,6,7}. The Cenozoic radiation of placental mammals, the foundational example of this concept^8,9, gave rise to much of the morphological disparity of extant mammals, and is generally attributed to relaxed evolutionary constraints following the extinction of non-avian dinosaurs^10,11,12,13. However, study of this and other radiations has focussed on variation in evolutionary rates^4,5,7,14, leaving the extent to which relaxation of constraints enabled the origin of novel phenotypes less well-characterised^15-17. We evaluate constraints on morphological evolution among mammaliaforms (mammals and their closest relatives) using a new method that quantifies the capacity of evolutionary change to generate phenotypic novelty. We find that Mesozoic crown-group therians, which include the ancestors of placental mammals, were significantly more constrained than other mammaliaforms. Relaxation of these constraints occurred in the mid-Paleocene, post-dating the extinction of non-avian dinosaurs at the K/Pg boundary, instead coinciding with important environmental shifts and with declining ecomorphological diversity in non-theriimorph mammaliaforms. This relaxation occurred even in small-bodied Cenozoic mammals weighing <100g, which are unlikely to have competed with dinosaurs. Instead, our findings support a more complex model whereby Mesozoic crown therian evolution was in-part constrained by co-occurrence with disparate mammaliaforms, as well as by presence of dinosaurs, within-lineage incumbency effects and environmental factors. Our results demonstrate that variation in evolutionary constraints can occur independently of variation in evolutionary rate; and that both make important contributions to the understanding of adaptive radiations.