Time-scaled phylogenetic trees are both an ultimate goal of evolutionary biology and a necessary ingredient in comparative studies. While accumulating genomic data has moved the field closer to a full description of the tree of life, the relative timing of certain evolutionary events remains challenging even when this data is abundant, and absolute timing is impossible without external information such as fossil ages and morphology. The field of phylogenetics lacks efficient tools integrating probabilistic models for these kinds of data into unified frameworks for estimating phylogenies. Here, we implement, benchmark and validate popular phylogenetic models for the study of paleontological and neontological continuous trait data, incorporating these models into the BEAST2 platform. Our methods scale well with number of taxa and of characters. We tip-date and estimate the topology of a phylogeny of Carnivora, comparing results from different configurations of integrative models capable of leveraging ages, as well as molecular and continuous morphological data from living and extinct species. Our results illustrate and advance the paradigm of Bayesian, probabilistic total evidence, in which explanatory models are fully defined, and inferential uncertainty in all their dimensions is accounted for.