Thanks for the pointers. This link presents a simple introduction to the issue: https://docs.pymc.io/notebooks/updating_priors.html
I would have expected that there was some discussion of this in the literature, though, since it seems like an important problem for practical application of MCMC and Bayesian methods.
So far, I have just been pretending that my distributions are conjugate, and updating the priors from the means and standard deviations, but I know that’s not right in general.