Flatness is stable under composition

Fix $x\in X$ and put $y=g(x)$, $s=f(y)$. Choose an affine open $U=\mathrm{Spec}A\subseteq S$ containing $s$; then $f^{-1}(U)$ is an open neighbourhood of $y$, so choose an affine open $V=\mathrm{Spec}B\subseteq f^{-1}(U)$ containing $y$; then $g^{-1}(V)$ is an open neighbourhood of $x$, so choose an affine open $W=\mathrm{Spec}C\subseteq g^{-1}(V)$ containing $x$. The charts give ring maps $A\to B\to C$ with $f(W)\subseteq V\subseteq U$.

Let $\mathfrak{n}$ be the prime of $W$ defining $x$, $\mathfrak{m}=\mathfrak{n}\cap B$ the prime of $V$ defining $y$, and $\mathfrak{p}=\mathfrak{m}\cap A$ the prime of $U$ defining $s$. Since $g$ is flat at $x$, [F2] gives $C_{\mathfrak{n}}$ flat over $B_{\mathfrak{m}}$; since $f$ is flat at $y$, [F2] gives $B_{\mathfrak{m}}$ flat over $A_{\mathfrak{p}}$. The localisation $C_{\mathfrak{n}}$ is also the localisation of the $B_{\mathfrak{m}}$-module $C{\otimes }_B{B}_{\mathfrak{m}}$ at $\mathfrak{n}$, so it is a flat $B_{\mathfrak{m}}$-module, and [F3] applied to the composite of flat ring maps $A_{\mathfrak{p}}\to B_{\mathfrak{m}}\to C_{\mathfrak{n}}$ makes $C_{\mathfrak{n}}$ flat over $A_{\mathfrak{p}}$.

The affine charts $W=\mathrm{Spec}C$ over $U=\mathrm{Spec}A$ satisfy $(f\circ g)(W)\subseteq U$, and step 2.1 exhibits flatness of $C_{\mathfrak{n}}$ over $A_{\mathfrak{p}}$, so [F2] gives that $f\circ g$ is flat at the arbitrary point $x$. By [F1] the composite $f\circ g$ is flat.

Degenerate cases are consistent: if $X$ is empty the composite has empty source and is flat vacuously; if $f$ or $g$ has empty source the pointwise hypotheses are vacuous; if both are identities the composite is the identity, and step 2.1 reads that $A_{\mathfrak{p}}$ is flat over itself. The argument makes no choice principle available or necessary, fixing one point and one nested chain of charts at a time. [F1, F2, step 2.1] $\mathrm{\square }$