The functor $F_n$ is a sheaf for the fppf topology by Lemma \ref{lemma-flat-dimension-n}. Observe that a monomorphism of finite presentation is separated and quasi-finite (Morphisms, Lemma \ref{morphisms-lemma-monomorphism-loc-finite-type-loc-quasi-finite}). Hence combining Descent, Lemma \ref{descent-lemma-descent-data-sheaves}, More on Morphisms, Lemma \ref{more-morphisms-lemma-separated-locally-quasi-finite-morphisms-fppf-descend} , and Descent, Lemmas \ref{descent-lemma-descending-property-monomorphism} and \ref{descent-lemma-descending-property-finite-presentation} we see that the question is local for the \'etale topology on $S$. \medskip\noindent In particular the situation is local for the Zariski topology on $S$ and we may assume that $S$ is affine. In this case the dimension of the fibres of $f$ is bounded above, hence we see that $F_n$ is representable for $n$ large enough. Thus we may use descending induction on $n$. Suppose that we know $F_{n + 1}$ is representable by a monomorphism $Z_{n + 1} \to S$ of finite presentation. Consider the base change $X_{n + 1} = Z_{n + 1} \times_S X$ and the pullback $\mathcal{F}_{n + 1}$ of $\mathcal{F}$ to $X_{n + 1}$. The morphism $Z_{n + 1} \to S$ is quasi-finite as it is a monomorphism of finite presentation, hence Lemma \ref{lemma-quasi-finite-base-change} implies that $\mathcal{F}_{n + 1}$ is pure relative to $Z_{n + 1}$. Since $F_n$ is a subfunctor of $F_{n + 1}$ we conclude that in order to prove the result for $F_n$ it suffices to prove the result for the corresponding functor for the situation $\mathcal{F}_{n + 1}/X_{n + 1}/Z_{n + 1}$. In this way we reduce to proving the result for $F_n$ in case $S_{n + 1} = S$, i.e., we may assume that $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over $S$. \medskip\noindent Fix $n$ and assume $\mathcal{F}$ is flat in dimensions $\geq n + 1$ over $S$. To finish the proof we have to show that $F_n$ is representable by a monomorphism $Z_n \to S$ of finite presentation. Since the question is local in the \'etale topology on $S$ it suffices to show that for every $s \in S$ there exists an elementary \'etale neighbourhood $(S', s') \to (S, s)$ such that the result holds after base change to $S'$. Thus by Lemma \ref{lemma-existence-complete} we may assume there exist \'etale morphisms $h_j : Y_j \to X$, $j = 1, \ldots, m$ such that for each $j$ there exists a complete d\'evissage of $\mathcal{F}_j/Y_j/S$ over $s$, where $\mathcal{F}_j$ is the pullback of $\mathcal{F}$ to $Y_j$ and such that $X_s \subset \bigcup h_j(Y_j)$. Note that by Lemma \ref{lemma-localize-flat-dimension-n} the sheaves $\mathcal{F}_j$ are still flat over in dimensions $\geq n + 1$ over $S$. Set $W = \bigcup h_j(Y_j)$, which is a quasi-compact open of $X$. As $\mathcal{F}$ is pure along $X_s$ we see that $$ E = \{t \in S \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset W \}. $$ contains all generalizations of $s$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-relative-assassin-constructible} $E$ is a constructible subset of $S$. We have seen that $\Spec(\mathcal{O}_{S, s}) \subset E$. By Morphisms, Lemma \ref{morphisms-lemma-constructible-containing-open} we see that $E$ contains an open neighbourhood of $s$. Hence after shrinking $S$ we may assume that $E = S$. It follows from Lemma \ref{lemma-localize-flat-dimension-n} that it suffices to prove the lemma for the functor $F_n$ associated to $X = \coprod Y_j$ and $\mathcal{F} = \coprod \mathcal{F}_j$. If $F_{j, n}$ denotes the functor for $Y_j \to S$ and the sheaf $\mathcal{F}_i$ we see that $F_n = \prod F_{j, n}$. Hence it suffices to prove each $F_{j, n}$ is representable by some monomorphism $Z_{j, n} \to S$ of finite presentation, since then $$ Z_n = Z_{1, n} \times_S \ldots \times_S Z_{m, n} $$ Thus we have reduced the theorem to the special case handled in Lemma \ref{lemma-flat-dimension-n-representable}.