Why is every elliptic curve over a proper base over $\mathbf{F}_q$ isotrivial? If the moduli space of elliptic curves $\mathbf{A}^1_\mathbf{Z}$ were fine, it would be clear to me.

It probably follows by considering the functor $\mathcal{M}_{1,1} \to \mathbf{A}^1_\mathbf{Z}$ in some way.