Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.

I feel that I should already know the answer to this, but it never sits quite right in my head.

When dealing with Gromov-Witten theory, one is obviously interested in part in the moduli stack of stable curves, $\overline{M}_{g,n}$ (actually, we probably should really care about the stack of pre-stable curves, but that's not going to be hugely relevant for my case).

When dealing with orbifold Gromov-Witten theory, we instead need to consider the moduli stack of twisted stable curves, which are those curves that may have isotropy at the marked points and nodes. For the problems that I am interested in, it is usually the case that every marked point has the same isotropy---in particular, this is $\mathbb{Z}/2$. Let us denote then the stack of twisted stable curves whose marked points all have $\mathbb{Z}/r$ isotropy as $\overline{M}_{g,n}^{\mathbb{Z}/r}$.

There is an obvious map $\overline{M}_{g,n}^{\mathbb{Z}/r} \to \overline{M}_{g,n}$ given by taking the coarse moduli space.

My question is: What is the relationship between these two stacks?

Edit further: Some further questions that I feel that are relevant.

The moduli spaces $\overline{M}_{g,n}$ have a lot of structure maps between them. For example, they have forgetful maps $\overline{M}_{g,n+1} \to \overline{M}_{g,n}$, gluing maps, etc. They also have a bunch of canonically defined bundles over them, such as the cotangent line bundles, the Hodge bundles, etc. In particular, the divisor classes of the cotangent line bundles can be computed recursively given a few base cases by pulling them back via the forgetful maps.

How does these work for the moduli spaces $\overline{M}_{g,n}^{\mathbb{Z}/r}$? From what I understand, the $\psi$-classes come from the coarse curve, and so all of the expected relations should still hold. In particular, for $g = 0$ we can describe the $\psi$-classes purely combinatorially in terms of partitioning the marked points among different components of our curve. Is this picture correct?

Perhaps a better question: Is there a good reference which covers this?