Let $G$ be a fixed finite group. I'm interested in the structure of the set $\mathcal{H}_{r,g,h,G}$ of tuples $(C,f,\delta)$, where $C$ is a smooth projective genus $g\geq 2$ curve, $\delta:G\to\mbox{Aut}(C)$ is an injective group homomorphism, $f:C\to C'$ is a finite Galois morphism with $r$ ramification points and Galois group $\delta(G)$, and $C'$ is a smooth projective curve of genus $h\geq 1$. When $h=0$, then this is just the usual Hurwitz scheme (or at least one of the usual ones) of Galois morphisms to $\mathbb{P}^1$ with fixed Galois group, and can be seen as the coarse moduli scheme of a certain functor (basically the above construction but relativized).

However, it seems to me that for $h>0$, not much is known. I have seen certain functors defined but where the $C'$ above is a fixed curve, not one that moves. I, on the other hand, am interested in not fixing the $C'$. Is there any literature on this? Can this set be seen as a coarse moduli scheme for a functor?