Let $M_{g} \to \mathbb{Z}$ be the coarse moduli scheme of non-singular genus g curves over $\mathbb{Z}$. That is, suppose that $M_{g}$ co-represents the functor $M^{\sharp}_{g} : \text{Sch} \to \text{Sets}$ whose Yoneda points are (flat) families of non-singular, genus $g$ curves.

What is $M_{g} \times \mathbb{Z}/p$?

One would expect that $M_{g} \times \mathbb{Z}/p$ co-represents $M^{\sharp}_{g} \times \mathbb{Z}/p$. In general, the formation of GIT quotients can fail to commute with passing to fibers. Does that failure occur here?

In other words: does $M_{g} \times \mathbb{Z}/p$ co-represent $M^{\sharp}_{g} \times \mathbb{Z}/p$?

Some references for the construction of $M_g$ over $\mathbb{Z}$ can be found here.