Families of covers give a space that we call a Hurwitz space. It's definition comes foremost from two well-known ideas: 1. It is defined as a cover of a configuration space through the use of the Hurwitz monodromy group. 2. That space is actually an algebraic variety from  (just Riemann's version of) Riemann's Existence Theorem (it does not need the famous Grauert-Remmert generalization).

It takes some doing to acquaint yourself with the use of the Hurwitz monodromy group (a combinatorial quotient of the Artin Braid group) and the permutation representations that produce various versions of Hurwitz spaces, depending on the equivalence one uses on covers. Still, as one's skill grows on that, it becomes a version of using Galois Theory. Permutation representations correspond to extensions (cover extensions instead of field extensions, though underlying it are field extensions) because of RET. I did that work a long time ago and I show you example results that now are documented in many places including two books, and papers in prestigious journals.