In terms of the free energy 'principle', it makes no predictions about how free energy minimized. But there have been multiple process theories suggested, most notably predictive coding (which is a dominant paradigm in neuroscience) [1] and variational message passing [2].

[1] https://en.wikipedia.org/wiki/Predictive_coding [2] http://www.jmlr.org/papers/volume6/winn05a/winn05a.pdf

Isn't variational message-passing the algorithmic-level theory about where predictive coding comes from?

I think you might be right, a quote from Friston on the relationship (in reference to belief propagation):

"We turn to the equivalent message passing for continuous variables, which transpires to be predictive coding [...]"

It could be that belief propagation is in the context of discrete variables, whereas predictive coding is in the context of continuous, both of which are a form of (variational) message passing.