Flatness is stable under arbitrary base change

Fix a point $x^{\mathrm{\prime }}\in X_{S^{\mathrm{\prime }}}=X{\times }_S{S}^{\mathrm{\prime }}$ and let $x\in X$, $s^{\mathrm{\prime }}\in S^{\mathrm{\prime }}$ be its images, $s=h(s^{\mathrm{\prime }})=f(x)$. Choose affine opens $U=\mathrm{Spec}A\subseteq S$ containing $s$, $V=\mathrm{Spec}{A}^{\mathrm{\prime }}\subseteq S^{\mathrm{\prime }}$ containing $s^{\mathrm{\prime }}$ with $h(V)\subseteq U$, and $W=\mathrm{Spec}B\subseteq X$ affine containing $x$ with $f(W)\subseteq U$. Then $W{\times }_{U}V={\mathrm{pr}}_X^{-1}(W)\cap {\mathrm{pr}}_{S^{\mathrm{\prime }}}^{-1}(V)$ is an affine open neighbourhood of $x^{\mathrm{\prime }}$ in $X_{S^{\mathrm{\prime }}}$, isomorphic to $\mathrm{Spec}(B{\otimes }_A{A}^{\mathrm{\prime }})$ by [F3], and it lies over $V$.

In the global case, $f$ is flat at every point of $W$, so the AC-qualified global converse in [F2], licensed by [F8], gives that $B$ is flat over $A$. Therefore [F7] gives that $B{\otimes }_A{A}^{\mathrm{\prime }}$ is flat over $A^{\mathrm{\prime }}$, and its localization at the prime of $x^{\mathrm{\prime }}$ is flat over $A_{{\mathfrak{p}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$. For the pointwise clause, [F2] gives only that $B_{\mathfrak{q}}$ is flat over $A_{\mathfrak{p}}$; no global flatness of $B$ is inferred from this local hypothesis.

Claim: $B{\otimes }_A{A}^{\mathrm{\prime }}$ is flat over $A^{\mathrm{\prime }}$. By [F5] it suffices to check injections $K\hookrightarrow N$ of $A^{\mathrm{\prime }}$-modules. Viewing $K,N$ as $A$-modules through $A\to A^{\mathrm{\prime }}$, associativity [F4] gives canonical isomorphisms $K{\otimes }_{A^{\mathrm{\prime }}}(B{\otimes }_A{A}^{\mathrm{\prime }})\cong K{\otimes }_{A}B$ and $N{\otimes }_{A^{\mathrm{\prime }}}(B{\otimes }_A{A}^{\mathrm{\prime }})\cong N{\otimes }_{A}B$ under which the induced map is $K{\otimes }_{A}B\to N{\otimes }_{A}B$; this is injective because $B$ is flat over $A$ and $K\hookrightarrow N$ is an injection of $A$-modules, by [F5]. Hence $B{\otimes }_A{A}^{\mathrm{\prime }}$ is flat over $A^{\mathrm{\prime }}$.

Applying [F2] to the affine charts $W{\times }_{U}V=\mathrm{Spec}(B{\otimes }_A{A}^{\mathrm{\prime }})$ over $V=\mathrm{Spec}{A}^{\mathrm{\prime }}$ converts the flatness of step 1.3 into flatness of $f_{S^{\mathrm{\prime }}}$ at the arbitrary point $x^{\mathrm{\prime }}$; by [F1] the base change is therefore flat, and [F6] identifies it as the pullback of $f$ along $h$. For the pointwise assertion, let $\mathfrak{r}$ be the prime of $B{\otimes }_A{A}^{\mathrm{\prime }}$ corresponding to $x^{\mathrm{\prime }}$, let $\mathfrak{q}$ be its contraction to $B$, and let ${\mathfrak{p}}^{\mathrm{\prime }}=\mathfrak{r}\cap A^{\mathrm{\prime }}$; set $\mathfrak{p}=\mathfrak{q}\cap A$. The assumption that $x^{\mathrm{\prime }}$ lies over $x$ means that $\mathfrak{q}$ is the prime of $x$. By [F2], $B_{\mathfrak{q}}$ is flat over $A_{\mathfrak{p}}$. Base-changing along $A_{\mathfrak{p}}\to A_{{\mathfrak{p}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$ and using [F7], $B_{\mathfrak{q}}{\otimes }_{A_{\mathfrak{p}}}{A}_{{\mathfrak{p}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$ is flat over $A_{{\mathfrak{p}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$. Localizing this algebra at the prime induced by $\mathfrak{r}$ gives $(B{\otimes }_A{A}^{\mathrm{\prime }}{)}_{\mathfrak{r}}$, which remains flat over $A_{{\mathfrak{p}}^{\mathrm{\prime }}}^{\mathrm{\prime }}$ by [F7]. The pointwise criterion [F2] now proves that $f_{S^{\mathrm{\prime }}}$ is flat at $x^{\mathrm{\prime }}$. This applies to every point over $x$. [F1, F2, F6, F7, step 1.3] $\mathrm{\square }$