A preliminary remark is that $X$, $S$ are affine schemes and that it suffices to prove $F_d$ is representable by a monomorphism of finite presentation $Z_d \to S$ on the category of affine schemes over $S$. (Of course we do not require $Z_d$ to be affine.) Hence throughout the proof of the lemma we work in the category of affine schemes over $S$. \medskip\noindent Let $(Z_k, Y_k, i_k, \pi_k, \mathcal{G}_k, \alpha_k)_{k = 1, \ldots, n}$ be a complete d\'evissage of $\mathcal{F}/X/S$ over $s$, see Definition \ref{definition-complete-devissage}. We will use induction on the length $n$ of the d\'evissage. Recall that $Y_k \to S$ is smooth with geometrically irreducible fibres, see Definition \ref{definition-one-step-devissage}. Let $d_k$ be the relative dimension of $Y_k$ over $S$. Recall that $i_{k, *}\mathcal{G}_k = \Coker(\alpha_k)$ and that $i_k$ is a closed immersion. By the definitions referenced above we have $d_1 = \dim(\text{Supp}(\mathcal{F}_s))$ and $$ d_k = \dim(\text{Supp}(\Coker(\alpha_{k - 1})_s)) = \dim(\text{Supp}(\mathcal{G}_{k, s})) $$ for $k = 2, \ldots, n$. It follows that $d_1 > d_2 > \ldots > d_n \geq 0$ because $\alpha_k$ is an isomorphism in the generic point of $(Y_k)_s$. \medskip\noindent Note that $i_1$ is a closed immersion and $\mathcal{F} = i_{1, *}\mathcal{G}_1$. Hence for any morphism of schemes $T \to S$ with $T$ affine, we have $\mathcal{F}_T = i_{1, T, *}\mathcal{G}_{1, T}$ and $i_{1, T}$ is still a closed immersion of schemes over $T$. Thus $\mathcal{F}_T$ is flat in dimensions $\geq d$ over $T$ if and only if $\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$. Because $\pi_1 : Z_1 \to Y_1$ is finite we see in the same manner that $\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$ if and only if $\pi_{1, T, *}\mathcal{G}_{1, T}$ is flat in dimensions $\geq d$ over $T$. The same arguments work for ``flat in dimensions $\geq d + 1$'' and we conclude in particular that $\pi_{1, *}\mathcal{G}_1$ is flat over $S$ in dimensions $\geq d + 1$ by our assumption on $\mathcal{F}$. \medskip\noindent Suppose that $d_1 > d$. It follows from the discussion above that in particular $\pi_{1, *}\mathcal{G}_1$ is flat over $S$ at the generic point of $(Y_1)_s$. By Lemma \ref{lemma-induction-step-fp} we may replace $S$ by an affine neighbourhood of $s$ and assume that $\alpha_1$ is $S$-universally injective. Because $\alpha_1$ is $S$-universally injective, for any morphism $T \to S$ with $T$ affine, we have a short exact sequence $$ 0 \to \mathcal{O}_{Y_{1, T}}^{\oplus r_1} \to \pi_{1, T, *}\mathcal{G}_{1, T} \to \Coker(\alpha_1)_T \to 0 $$ and still the first arrow is $T$-universally injective. Hence the set of points of $(Y_1)_T$ where $\pi_{1, T, *}\mathcal{G}_{1, T}$ is flat over $T$ is the same as the set of points of $(Y_1)_T$ where $\Coker(\alpha_1)_T$ is flat over $S$. In this way the question reduces to the sheaf $\Coker(\alpha_1)$ which has a complete d\'evissage of length $n - 1$ and we win by induction. \medskip\noindent If $d_1 < d$ then $F_d$ is represented by $S$ and we win. \medskip\noindent The last case is the case $d_1 = d$. This case follows from a combination of Lemma \ref{lemma-compare-H-F} and Lemma \ref{lemma-free-at-generic-points-representable}.