The Hurwitz action of braid groups and the computational complexity of knot invariants

Fix a finite group $G$, and a conjugacy invariant subset $C$ of $G$. The $n$-strand braid group $B_n$ acts on the Cartesian product $C^n$, which we can think of as the set of regular branched $G$-covers of the disk with branch types in C. What are the orbits of this action?
I’ll answer this question in the stable range where $n$ is "large enough," under the assumption that $C$ generates $G$. More generally, I will answer analogous questions for surfaces with both punctures and genus, without requiring that $C$ generate $G$. This provides a mutual generalization of theorems of Ellenberg-Venkatesh-Westerland and Dunfield-Thurston.
When $G$ is assumed to be a non-abelian simple group, we can say much more about the stable behavior of these actions. In particular, in joint work with Greg Kuperberg, we used this understanding to derive complexity-theoretic hardness results for $G$-coloring invariants of knots.