Hi Peter, thanks for this answer! Since asking this question I've gone more into equivariant homotopy theory, and recently have been thinking about connections between that and prisms. I wrote up my thoughts in a separate answer.

@KConrad there's no restriction on the $k_i$. For the combinatorial interpretation I'd want to know what happens when $q=1$ and when $q=p^m$, but for my applications $q-1$ is going to specialize to $\mu\in \mathbf A_{\mathrm{inf}}$; see the motivation added to the question.