As requested, I am posting my comment as an answer. Mumford discusses this in the preface to the first edition of "Geometric Invariant Theory".

Already in "Geometric Invariant Theory" (which was Mumford's thesis), Mumford constructed the coarse moduli scheme $M_g$ over $\text{Spec}\ \mathbb{Z}.$ You can read about what this means in Mumford's book. The great irony of GIT is that, although GIT was largely motivated by this problem, in fact Mumford constructs $M_g$ over $\text{Spec}\ \mathbb{Z}$ without using GIT (precisely because neither had Haboush yet extended GIT to char $p$, nor had Seshadri extended GIT to mixed characteristic).