Let $S$ be a connected scheme, let $\pi : \mathbb{P}_{S}^{r} \to S$ be projective $r$-space over $S$, and let $\mathcal{E}$ be a flat and locally finitely presented $\mathcal{O}_{\mathbb{P}_{S}^{r}}$-module. Is $\pi_{\ast}(\mathcal{E}(n))$ (nonzero and) flat and locally finitely presented for $n \gg 0$?

Remarks: If $S$ is a Noetherian scheme, the answer is "yes" using usual cohomology and base change theorems. I included "connected" because I am assuming we could otherwise construct an example with $S$ being an infinite disjoint union of fields. The case when $S$ is quasi-compact reduces to the case when $S$ is affine, but I don't even know about the affine case. Say $S = \operatorname{Spec} A$; we may write $A$ as a filtered colimit $A = \varinjlim_{\lambda \in \Lambda} A_{\lambda}$ where each $A_{\lambda}$ is a finite type $\mathbb{Z}$-subalgebra of $A$. Set $S_{\lambda} := \operatorname{Spec} A_{\lambda}$. Then we can descend $\mathcal{E}$ to vector bundles $\mathcal{E}_{\lambda}$ on $\mathbb{P}_{S_{\lambda}}^{r}$ for large enough $\lambda$. By the case when $S$ is Noetherian, we'd be done if $\Gamma(\mathbb{P}_{S_{\lambda}}^{r},\mathcal{E}_{\lambda}) \otimes_{A_{\lambda}} A \to \Gamma(\mathbb{P}_{S}^{r} , \mathcal{E})$ is an isomorphism for some $\lambda$. By Tag 01Z0, the map $\varinjlim_{\lambda \in \Lambda} \Gamma(\mathbb{P}_{S_{\lambda}}^{r} , \mathcal{E}_{\lambda}) \to \Gamma(\mathbb{P}_{S}^{r} , \mathcal{E})$ is an isomorphism but I don't know if this helps.

I am guessing that the difficulty is "flatness" but I also don't know whether the "locally finitely presented" part is true, thus I will include the following subquestion:

Let $S$ be an affine scheme, let $\pi : \mathbb{P}_{S}^{r} \to S$ be projective $r$-space over $S$, and let $\mathcal{E}$ be a locally finitely presented $\mathcal{O}_{\mathbb{P}_{S}^{r}}$-module. Is $\pi_{\ast}\mathcal{E}$ necessarily finitely presented?

Keywords: Noetherian approximation, cohomology and base change, higher direct images, not Noetherian