Suppose $X \to S$ is a proper (projective) morphism of schemes, $S$ is quasi-compact and quasi-separated, and $\mathcal L$ is a relatively ample sheaf on $X$. By stacks, tag 01ZA, the scheme $S$ is a limit of a directed system of schemes $(S_i, f_{ij})$, where each $S_i$ is of finite type over $\mathbb Z$, and each $f_{ij}$ is affine. By stacks, tag 01ZM, there exists an index $i \in I$ and a scheme $X_i \to S_i$, such that $X$ is the pull-back of $X$ along $S \to S_i$. Similarly, by tag 01ZR, we can choose $i$ such that there is also a coherent sheaf $\mathcal L_i$ on $X_i$ such that $\mathcal L$ is the pull-back of $\mathcal L_i$. By tag 081F we may even assume that $X_i \to S_i$ is proper.
I wonder if we may also assume that $\mathcal L_i$ is ample?
I think this is true. So as $S$ is quasi-compact, some power $\mathcal L^{\otimes n}$ will be relatively very ample, inducing a closed immersion $$j: X \hookrightarrow \mathbf P(E).$$ Then we may descend both $X$ and $\mathbf P(E)$ to a closed immersion $X_i \subset \mathbf P(E_i)$. We may also descend the isomorphism $\mathcal L^{\otimes n} \cong j^* \mathcal O(1)$ to an isomorphism $\mathcal L_i^{\otimes n} \cong j_i^* \mathcal O(1)$, which shows that $\mathcal L_i$ is ample.