Factors that influence the distribution, abundance, and diversification of species can simultaneously affect multiple evolutionary lineages within or across communities. These include environmental changes and inter-specific ecological interactions that cause ranges of multiple, co-distributed species to contract, expand, or become fragmented. Such processes predict genetic patterns consistent with temporally clustered evolutionary events across species, such as synchronous population divergences and/or changes in population size. There have been a number of methods developed to infer shared divergences or changes in effective population size, but not both, and the latter has been limited to approximate Bayesian computation (ABC). We introduce a general, full-likelihood Bayesian method that can use genomic data to estimate temporal clustering of an arbitrary mix of population divergences and populationsize changes across taxa. Applying this method to simulated data, we find that estimating the timing and sharing of demographic changes is much more challenging than divergences. Even under favorable simulation conditions, the ability to infer shared demographic events is quite limited and very sensitive to prior assumptions, which is in sharp contrast to accurate, precise, and robust estimates of shared divergence times. Our results also suggest that previous estimates of co-expansion among five Alaskan populations of threespine sticklebacks (Gasterosteus aculeatus) were likely driven by a combination of prior assumptions and the lack of information about the timing of demographic changes when invariant characters are ignored. We conclude by discussing potential avenues to improve the estimation of synchronous demographic changes across populations.