ag.algebraic geometry - Coarse moduli spaces over Z and F_p

I would like to know to what extent it is possible to compare fibers over $mathbb{F}_p$ of coarse moduli spaces over $mathbb{Z}$, and coarse moduli spaces over $mathbb{F}_p$. I ask a more precise question below.

Let $mathcal{M}_g^{mathbb{Z}}$ be the moduli stack of smooth genus $g$ curves over $mathbb{Z}$. Let $M_g^{mathbb{Z}}$ be its coarse moduli space, and $(M_g^{mathbb{Z}})_p$ the fiber of this coarse moduli space over $mathbb{F}_p$.
Let $mathcal{M}_g^{mathbb{F}_p}$ be the moduli stack of smooth genus $g$ curves over $mathbb{F}_p$ and $M_g^{mathbb{F}_p}$ its coarse moduli space.

The universal property gives a map $phi:M_g^{mathbb{F}_p}rightarrow(M_g^{mathbb{Z}})_p$. My question is : is $phi$ an isomorphism ?

In fact, since $phi$ is a bijection between geometric points, and $M_g^{mathbb{F}_p}$ is normal, the question can be reformulated as : is $(M_g^{mathbb{Z}})_p$ normal ? This shows that when $g$ is fixed, the answer is "yes" except for a finite number of primes $p$.

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