Let $y \in Y$ be a point over $s$. It suffices to find an open neighbourhood of $y$ with the desired properties. As a first step, we replace $Y$ by the open $V$ found in Lemma \ref{lemma-h1-fibre-zero} so that $R^1f_*\mathcal{O}_X$ is zero universally (the hypothesis of the lemma holds by Lemma \ref{lemma-check-h1-fibre-zero}). We also shrink $Y$ so that all fibres of $f$ have dimension $\leq 1$ (use Morphisms, Lemma \ref{morphisms-lemma-openness-bounded-dimension-fibres} and properness of $f$). Thus we may assume we have (b) and (c) with $V = Y$ and after any base change $Y' \to Y$. Thus by Lemma \ref{lemma-h1-fibre-zero-check-h0-kappa} it now suffices to show (d) over $Y$. We may still shrink $Y$ further; for example, we may and do assume $Y$ and $S$ are affine. \medskip\noindent By Theorem \ref{theorem-openness-flatness} there is an open subset $U \subset X$ where $X \to S$ is flat which contains $X_s$ by hypothesis. Then $f(X \setminus U)$ is a closed subset not containing $y$. Thus after shrinking $Y$ we may assume $X$ is flat over $S$. \medskip\noindent Say $S = \Spec(R)$. Choose a closed immersion $Y \to Y'$ where $Y'$ is the spectrum of a polynomial ring $R[x_e; e \in E]$ on a set $E$. Denote $f' : X \to Y'$ the composition of $f$ with $Y \to Y'$. Then the hypotheses (1) -- (4) as well as (b) and (c) hold for $f'$ and $s$. If we we show $\mathcal{O}_{Y'} \to f'_*\mathcal{O}_X$ is surjective in an open neighbourhood of $y$, then the same is true for $\mathcal{O}_Y \to f_*\mathcal{O}_X$. Thus we may assume $Y$ is the spectrum of $R[x_e; e \in E]$. \medskip\noindent At this point $X$ and $Y$ are flat over $S$. Then $Y_s$ and $X$ are tor independent over $Y$. We urge the reader to find their own proof, but it also follows from Lemma \ref{lemma-case-of-tor-independence} applied to the square with corners $X, Y, S, S$ and its base change by $s \to S$. Hence $$ Rf_{s, *}\mathcal{O}_{X_s} = L(Y_s \to Y)^*Rf_*\mathcal{O}_X $$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-compare-base-change}. Because of the vanishing already established this implies $f_{s, *}\mathcal{O}_{X_s} = (Y_s \to Y)^*f_*\mathcal{O}_X$. We conclude that $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is a map of quasi-coherent $\mathcal{O}_Y$-modules whose pullback to $Y_s$ is surjective. We claim $f_*\mathcal{O}_X$ is a finite type $\mathcal{O}_Y$-module. If true, then the cokernel $\mathcal{F}$ of $\mathcal{O}_Y \to f_*\mathcal{O}_X$ is a finite type quasi-coherent $\mathcal{O}_Y$-module such that $\mathcal{F}_y \otimes \kappa(y) = 0$. By Nakayama's lemma (Algebra, Lemma \ref{algebra-lemma-NAK}) we have $\mathcal{F}_y = 0$. Thus $\mathcal{F}$ is zero in an open neighbourhood of $y$ (Modules, Lemma \ref{modules-lemma-finite-type-stalk-zero}) and the proof is complete. \medskip\noindent Proof of the claim. For a finite subset $E' \subset E$ set $Y' = \Spec(R[x_e; e \in E'])$. For large enough $E'$ the morphism $f' : X \to Y \to Y'$ is proper, see Limits, Lemma \ref{limits-lemma-finite-type-eventually-proper}. We fix $E'$ and $Y'$ in the following. Write $R = \colim R_i$ as the colimit of its finite type $\mathbf{Z}$-subalgebras. Set $S_i = \Spec(R_i)$ and $Y'_i = \Spec(R_i[x_e; e \in E'])$. For $i$ large enough we can find a diagram $$ \xymatrix{ X \ar[d] \ar[r]_{f'} & Y' \ar[d] \ar[r] & S \ar[d] \\ X_i \ar[r]^{f'_i} & Y'_i \ar[r] & S_i } $$ with cartesian squares such that $X_i$ is flat over $S_i$ and $X_i \to Y'_i$ is proper. See Limits, Lemmas \ref{limits-lemma-descend-finite-presentation}, \ref{limits-lemma-descend-flat-finite-presentation}, and \ref{limits-lemma-eventually-proper}. The same argument as above shows $Y'$ and $X_i$ are tor independent over $Y'_i$ and hence $$ R\Gamma(X, \mathcal{O}_X) = R\Gamma(X_i, \mathcal{O}_{X_i}) \otimes^\mathbf{L}_{R_i[x_e; e \in E']} R[x_e; e \in E'] $$ by the same reference as above. By Cohomology of Schemes, Lemma \ref{coherent-lemma-proper-over-affine-cohomology-finite} the complex $R\Gamma(X_i, \mathcal{O}_{X_i})$ is pseudo-coherent in the derived category of the Noetherian ring $R_i[x_e; e \in E']$ (see More on Algebra, Lemma \ref{more-algebra-lemma-Noetherian-pseudo-coherent}). Hence $R\Gamma(X, \mathcal{O}_X)$ is pseudo-coherent in the derived category of $R[x_e; e \in E']$, see More on Algebra, Lemma \ref{more-algebra-lemma-pull-pseudo-coherent}. Since the only nonvanishing cohomology module is $H^0(X, \mathcal{O}_X)$ we conclude it is a finite $R[x_e; e \in E']$-module, see More on Algebra, Lemma \ref{more-algebra-lemma-n-pseudo-module}. This concludes the proof.