Mutation-selection phylogenetic codon models are grounded on population genetics first principles and represent a principled approach for investigating the intricate interplay between mutation, selection, and drift. In their current form, mutation-selection codon models are entirely characterized by the collection of site-specific amino-acid fitness profiles. However, thus far, they have relied on the assumption of a constant genetic drift, translating into a unique effective population size (Ne) across the phylogeny, clearly an unrealistic assumption. This assumption can be alleviated by introducing variation in Ne between lineages. In addition to Ne, the mutation rate (μ) is susceptible to vary between lineages, and both should covary with life-history traits (LHTs). This suggests that the model should more globally account for the joint evolutionary process followed by all of these lineage-specific variables (Ne, μ, and LHTs). In this direction, we introduce an extended mutation-selection model jointly reconstructing in a Bayesian Monte Carlo framework the fitness landscape across sites and long-term trends in Ne, μ, and LHTs along the phylogeny, from an alignment of DNA coding sequences and a matrix of observed LHTs in extant species. The model was tested against simulated data and applied to empirical data in mammals, isopods, and primates. The reconstructed history of Ne in these groups appears to correlate with LHTs or ecological variables in a way that suggests that the reconstruction is reasonable, at least in its global trends. On the other hand, the range of variation in Ne inferred across species is surprisingly narrow. This last point suggests that some of the assumptions of the model, in particular concerning the assumed absence of epistatic interactions between sites, are potentially problematic.