Let $U \subset X$ be the set of points where $\mathcal{F}$ is flat over $S$. By More on Morphisms, Theorem \ref{more-morphisms-theorem-openness-flatness} this is an open subscheme of $X$. By assumption $X_s \subset U$. As $X_s$ is quasi-compact, we can find a quasi-compact open $U' \subset U$ with $X_s \subset U'$. By Limits, Lemma \ref{limits-lemma-descend-finite-presentation} we can find an $i \in I$ and a morphism of finite presentation $f_i : X_i \to S_i$ whose base change to $S$ is isomorphic to $f_i$. Fix such a choice and set $X_{i'} = X_i \times_{S_i} S_{i'}$. Then $X = \lim_{i'} X_{i'}$ with affine transition morphisms. By Limits, Lemma \ref{limits-lemma-descend-modules-finite-presentation} we can, after possible increasing $i$ assume there exists a quasi-coherent $\mathcal{O}_{X_i}$-module $\mathcal{F}_i$ of finite presentation whose base change to $S$ is isomorphic to $\mathcal{F}$. By Limits, Lemma \ref{limits-lemma-descend-opens} after possibly increasing $i$ we may assume there exists an open $U'_i \subset X_i$ whose inverse image in $X$ is $U'$. Note that in particular $(X_i)_{s_i} \subset U'_i$. By Limits, Lemma \ref{limits-lemma-descend-module-flat-finite-presentation} (after increasing $i$ once more) we may assume that $\mathcal{F}_i$ is flat on $U'_i$. In particular we see that $\mathcal{F}_i$ is flat along $(X_i)_{s_i}$. \medskip\noindent Next, we use Lemma \ref{lemma-finite-presentation-flat-along-fibre} to choose an elementary \'etale neighbourhood $(S_i', s_i') \to (S_i, s_i)$ and a commutative diagram of schemes $$ \xymatrix{ X_i \ar[d] & X_i' \ar[l]^{g_i} \ar[d] \\ S_i & S_i' \ar[l] } $$ such that $g_i$ is \'etale, $(X_i)_{s_i} \subset g_i(X_i')$, the schemes $X_i'$, $S_i'$ are affine, and such that $\Gamma(X_i', g_i^*\mathcal{F}_i)$ is a projective $\Gamma(S_i', \mathcal{O}_{S_i'})$-module. Note that $g_i^*\mathcal{F}_i$ is universally pure over $S'_i$, see Lemma \ref{lemma-affine-locally-projective-pure}. We may base change the diagram above to a diagram with morphisms $(S'_{i'}, s'_{i'}) \to (S_{i'}, s_{i'})$ and $g_{i'} : X'_{i'} \to X_{i'}$ over $S_{i'}$ for any $i' \geq i$ and we may base change the diagram to a diagram with morphisms $(S', s') \to (S, s)$ and $g : X' \to X$ over $S$. \medskip\noindent At this point we can use our criterion for purity. Set $W'_i \subset X_i \times_{S_i} S'_i$ equal to the image of the \'etale morphism $X'_i \to X_i \times_{S_i} S'_i$. For every $i' \geq i$ we have similarly the image $W'_{i'} \subset X_{i'} \times_{S_{i'}} S'_{i'}$ and we have the image $W' \subset X \times_S S'$. Taking images commutes with base change, hence $W'_{i'} = W'_i \times_{S'_i} S'_{i'}$ and $W' = W_i \times_{S'_i} S'$. Because $\mathcal{F}$ is pure along $X_s$ the Lemma \ref{lemma-criterion} implies that \begin{equation} \label{equation-inclusion} f^{-1}(\Spec(\mathcal{O}_{S', s'})) \cap \text{Ass}_{X \times_S S'/S'}(\mathcal{F} \times_S S') \subset W' \end{equation} By More on Morphisms, Lemma \ref{more-morphisms-lemma-relative-assassin-constructible} we see that $$ E = \{t \in S' \mid \text{Ass}_{X_t}(\mathcal{F}_t) \subset W' \} \quad\text{and}\quad E_{i'} = \{t \in S'_{i'} \mid \text{Ass}_{X_t}(\mathcal{F}_{i', t}) \subset W'_{i'} \} $$ are locally constructible subsets of $S'$ and $S'_{i'}$. By More on Morphisms, Lemma \ref{more-morphisms-lemma-base-change-assassin-in-U} we see that $E_{i'}$ is the inverse image of $E_i$ under the morphism $S'_{i'} \to S'_i$ and that $E$ is the inverse image of $E_i$ under the morphism $S' \to S'_i$. Thus Equation (\ref{equation-inclusion}) is equivalent to the assertion that $\Spec(\mathcal{O}_{S', s'})$ maps into $E_i$. As $\mathcal{O}_{S', s'} = \colim_{i' \geq i} \mathcal{O}_{S'_{i'}, s'_{i'}}$ we see that $\Spec(\mathcal{O}_{S'_{i'}, s'_{i'}})$ maps into $E_i$ for some $i' \geq i$, see Limits, Lemma \ref{limits-lemma-limit-contained-in-constructible}. Then, applying Lemma \ref{lemma-criterion} to the situation over $S_{i'}$, we conclude that $\mathcal{F}_{i'}$ is pure along $(X_{i'})_{s_{i'}}$.