Let $S$ be a Noetherian scheme and $\pi_{S}:\mathbb{P}^{n}_{S}=\mathbb{P}^{n}_{\mathbb{Z}}\times S$ be the usual projection. Let $\mathcal{F}$ be a coherent sheaf over $\mathbb{P}^{n}_{S}$. Since the category of coherent sheaves has enough projectives, one can form the right-derived functor $R\pi_{S*}\mathcal{F}$. Nitsure (https://arxiv.org/abs/math/0504590, p.14) makes the claim that for large-enough $r$, the first right-derived $R^{1}\pi_{S*}\mathcal{F}(r)$ vanishes.
Why is that so? Is this somehow analogous to Serre's theorem that $H^{i}(X,\mathcal{F}(r))$ vanishes for large-enough $r$ and $i\geq 1$?
Your argument about the right derived functor is unfortunately incorrect. Projective schemes of positive dimension don't have enough projectives; the correct justification for the existence of the right derived functor is that there are enough injectives, see for instance Hartshorne section III.8.
Anyways, back to the issue at hand. The key fact here is the following:
With this in hand, we see that your conjecture is entirely correct. Cover $S$ with finitely many noetherian affine open subschemes $\operatorname{Spec} R_j$ and apply Serre's result to the induced maps $X\times_S \operatorname{Spec} R_j\to \operatorname{Spec} R_j$. Since there are finitely many $\operatorname{Spec} R_j$, we can pick some $r\gg 0$ so that $H^i(X\times_S \operatorname{Spec} R_j,\mathcal{F}|_\cdots(r))$ vanishes for all $j$. Since derived pushforward commutes with restriction on the target, we find that $R^if_*\mathcal{F}(r)=0$.