We will verify conditions (1), (2), (3), (4) of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Since $0$ is a quasi-coherent module on any ringed site we see that (1) holds. By definition $\QCoh(\mathcal{O})$ is a strictly full subcategory $\textit{Mod}(\mathcal{O})$, so (2) holds. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a morphism of quasi-coherent modules on $\mathcal{X}_{lisse,\etale}$ or $\mathcal{X}_{flat,fppf}$. We have $g^*g_!\mathcal{F} = \mathcal{F}$ and similarly for $\mathcal{G}$ and $\varphi$, see Lemma \ref{lemma-lisse-etale-modules}. By Lemma \ref{lemma-shriek-quasi-coherent} we see that $g_!\mathcal{F}$ and $g_!\mathcal{G}$ are quasi-coherent $\mathcal{O}_\mathcal{X}$-modules. Hence we see that $\Ker(g_!\varphi)$ and $\Coker(g_!\varphi)$ are quasi-coherent modules on $\mathcal{X}$. Since $g^*$ is exact (see Lemma \ref{lemma-lisse-etale}) we see that $g^*\Ker(g_!\varphi) = \Ker(g^*g_!\varphi) = \Ker(\varphi)$ and $g^*\Coker(g_!\varphi) = \Coker(g^*g_!\varphi) = \Coker(\varphi)$ are quasi-coherent too (see Lemma \ref{lemma-quasi-coherent}). This proves (3). \medskip\noindent Finally, suppose that $$ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0 $$ is an extension of $\mathcal{O}_{\mathcal{X}_{lisse,\etale}}$-modules (resp.\ $\mathcal{O}_{\mathcal{X}_{flat,fppf}}$-modules) with $\mathcal{F}$ and $\mathcal{G}$ quasi-coherent. We have to show that $\mathcal{E}$ is quasi-coherent on $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$). We strongly urge the reader to write out what this means on a napkin and prove it him/herself rather than reading the somewhat labyrinthine proof that follows. By Lemma \ref{lemma-quasi-coherent} this is true if and only if $g_!\mathcal{E}$ is quasi-coherent. By Lemmas \ref{lemma-check-qc-on-etale-covering} and Lemma \ref{lemma-lisse-etale-functorial} we may check this after replacing $\mathcal{X}$ by a smooth (resp.\ fppf) cover. Choose a scheme $U$ and a smooth surjective morphism $U \to \mathcal{X}$. By definition there exists an \'etale (resp.\ fppf) covering $\{U_i \to U\}_i$ such that $\mathcal{G}$ has a global presentation over each $U_i$. Replacing $\mathcal{X}$ by $U_i$ (which is permissible by the discussion above) we may assume that the site $\mathcal{X}_{lisse,\etale}$ (resp.\ $\mathcal{X}_{flat,fppf}$) has a final object $U$ (in other words $\mathcal{X}$ is representable by the scheme $U$) and that $\mathcal{G}$ has a global presentation $$ \bigoplus\nolimits_{j \in J} \mathcal{O} \longrightarrow \bigoplus\nolimits_{i \in I} \mathcal{O} \longrightarrow \mathcal{G} \longrightarrow 0 $$ of $\mathcal{O}$-modules where $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{lisse,\etale}}$ (resp.\ $\mathcal{O} = \mathcal{O}_{\mathcal{X}_{flat,fppf}}$). Let $\mathcal{E}'$ be the pullback of $\mathcal{E}$ via the map $\bigoplus\nolimits_{i \in I} \mathcal{O} \to \mathcal{G}$. Then we see that $\mathcal{E}$ is the cokernel of a map $\bigoplus\nolimits_{j \in J} \mathcal{O} \to \mathcal{E}'$ hence by property (3) which we proved above, it suffices to prove that $\mathcal{E}'$ is quasi-coherent. Consider the exact sequence $$ L_1g_!\left(\bigoplus\nolimits_{i \in I}\mathcal{O}\right) \to g_!\mathcal{F} \to g_!\mathcal{E}' \to g_!\left(\bigoplus\nolimits_{i \in I}\mathcal{O}\right) \to 0 $$ where $L_1g_!$ is the first left derived functor of $g_! : \textit{Mod}(\mathcal{O}_{\mathcal{X}_{lisse,\etale}}) \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ (resp.\ $g_! : \textit{Mod}(\mathcal{X}_{flat,fppf}, \mathcal{O}_{\mathcal{X}_{flat,fppf}}) \to \textit{Mod}(\mathcal{X}_{fppf}, \mathcal{O}_{\mathcal{X}})$). This derived functor exists by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-existence-derived-lower-shriek}. Moreover, since $\mathcal{O} = j_{U!}\mathcal{O}_U$ we have $Lg_!\mathcal{O} = g_!\mathcal{O} = \mathcal{O}_\mathcal{X}$ also by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-existence-derived-lower-shriek}. Thus the left hand term vanishes and we obtain a short exact sequence $$ 0 \to g_!\mathcal{F} \to g_!\mathcal{E}' \to \bigoplus\nolimits_{i \in I}\mathcal{O}_\mathcal{X} \to 0 $$ By Proposition \ref{proposition-lcq-flat-base-change} it follows that $g_!\mathcal{E}'$ is locally quasi-coherent with the flat base change property. Finally, Lemma \ref{lemma-quasi-coherent} implies that $\mathcal{E}' = g^{-1}g_!\mathcal{E}'$ is quasi-coherent as desired.