Proof of (1). Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{G}$ and think of $\mathcal{G}$ as a finite type quasi-coherent module on $Z$. We may replace $X$ by $Z$ and $u$ by the map $i^*\mathcal{F} \to \mathcal{G}$ (details omitted). Hence we may assume $f$ is quasi-compact and $\mathcal{G}$ of finite type. Let $T = \lim_{i \in I} T_i$ be a directed limit of affine $B$-schemes and assume that $u_T$ is surjective. Set $X_i = X_{T_i} = X \times_S T_i$ and $u_i = u_{T_i} : \mathcal{F}_i = \mathcal{F}_{T_i} \to \mathcal{G}_i = \mathcal{G}_{T_i}$. To prove (1) we have to show that $u_i$ is surjective for some $i$. Pick $0 \in I$ and replace $I$ by $\{i \mid i \geq 0\}$. Since $f$ is quasi-compact we see $X_0$ is quasi-compact. Hence we may choose a surjective \'etale morphism $\varphi_0 : W_0 \to X_0$ where $W_0$ is an affine scheme. Set $W = W_0 \times_{T_0} T$ and $W_i = W_0 \times_{T_0} T_i$ for $i \geq 0$.  These are affine schemes endowed with a surjective \'etale morphisms $\varphi : W \to X_T$ and $\varphi_i : W_i \to X_i$. Note that $W = \lim W_i$. Hence $\varphi^*u_T$ is surjective and it suffices to prove that $\varphi_i^*u_i$ is surjective for some $i$. Thus we have reduced the problem to the affine case which is Algebra, Lemma \ref{algebra-lemma-module-map-property-in-colimit} part (2). \medskip\noindent Proof of (2). Assume $\mathcal{F}$ is of finite type with scheme theoretic support $Z \subset B$ quasi-compact over $B$. Let $T = \lim_{i \in I} T_i$ be a directed limit of affine $B$-schemes and assume that $u_T$ is zero. Set $X_i = T_i \times_B X$ and denote $u_i : \mathcal{F}_i \to \mathcal{G}_i$ the pullback. Choose $0 \in I$ and replace $I$ by $\{i \mid i \geq 0\}$. Set $Z_0 = Z \times_X X_0$. By Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-support-finite-type} the support of $\mathcal{F}_i$ is $|Z_0|$. Since $|Z_0|$ is quasi-compact we can find an affine scheme $W_0$ and an \'etale morphism $W_0 \to X_0$ such that $|Z_0| \subset \Im(|W_0| \to |X_0|)$. Set $W = W_0 \times_{T_0} T$ and $W_i = W_0 \times_{T_0} T_i$ for $i \geq 0$. These are affine schemes endowed with \'etale morphisms $\varphi : W \to X_T$ and $\varphi_i : W_i \to X_i$. Note that $W = \lim W_i$ and that the support of $\mathcal{F}_T$ and $\mathcal{F}_i$ is contained in the image of $|W| \to |X_T|$ and $|W_i| \to |X_i|$. Now $\varphi^*u_T$ is injective and it suffices to prove that $\varphi_i^*u_i$ is injective for some $i$. Thus we have reduced the problem to the affine case which is Algebra, Lemma \ref{algebra-lemma-module-map-property-in-colimit} part (1). \medskip\noindent Proof of (3). This can be proven in exactly the same manner as in the previous two paragraphs using Algebra, Lemma \ref{algebra-lemma-module-map-property-in-colimit} part (3). We can also deduce it from (1) and (2) as follows. Let $T = \lim_{i \in I} T_i$ be a directed limit of affine $B$-schemes and assume that $u_T$ is an isomorphism. By part (1) there exists an $0 \in I$ such that $u_{T_0}$ is surjective. Set $\mathcal{K} = \Ker(u_{T_0})$ and consider the map of quasi-coherent modules $v : \mathcal{K} \to \mathcal{F}_{T_0}$. For $i \geq 0$ the base change $v_{T_i}$ is zero if and only if $u_i$ is an isomorphism. Moreover, $v_T$ is zero. Since $\mathcal{G}_{T_0}$ is of finite presentation, $\mathcal{F}_{T_0}$ is of finite type, and $u_{T_0}$ is surjective we conclude that $\mathcal{K}$ is of finite type (Modules on Sites, Lemma \ref{sites-modules-lemma-kernel-surjection-finite-onto-finite-presentation}). It is clear that the support of $\mathcal{K}$ is contained in the support of $\mathcal{F}_{T_0}$ which is quasi-compact over $T_0$. Hence we can apply part (2) to see that $v_{T_i}$ is zero for some $i$.