Let $g : \mathcal{Y} \to \mathcal{X}$ be as in (1). Let $y$ be an object of $\mathcal{Y}$ lying over a scheme $V$. By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-comparison} we have $(g^*\mathcal{F})|_{V_\etale} = \mathcal{F}|_{V_\etale}$. Moreover a comparison mapping for the sheaf $g^*\mathcal{F}$ on $\mathcal{Y}$ is a special case of a comparison map for the sheaf $\mathcal{F}$ on $\mathcal{X}$, see Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-comparison}. In this way (1) is clear. \medskip\noindent Proof of (2). We use the characterization of weak Serre subcategories of Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}. Kernels and cokernels of maps between sheaves having the flat base change property also have the flat base change property. This is clear because $f_{small}^*$ is exact for a flat morphism of schemes and since the restriction functors $(-)|_{U_\etale}$ are exact (because we are working in the \'etale topology). Finally, if $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is a short exact sequence of $\textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$ and the outer two sheaves have the flat base change property then the middle one does as well, again because of the exactness of $f_{small}^*$ and the restriction functors (and the 5 lemma). \medskip\noindent Proof of (3). Let $f_i : \mathcal{X}_i \to \mathcal{X}$ be a jointly surjective family of smooth morphisms of algebraic stacks and assume each $f_i^*\mathcal{F}$ has the flat base change property. By part (1), the definition of an algebraic stack, and the fact that compositions of smooth morphisms are smooth (see Morphisms of Stacks, Lemma \ref{stacks-morphisms-lemma-composition-smooth}) we may assume that each $\mathcal{X}_i$ is representable by a scheme. Let $\varphi : x \to x'$ be a morphism of $\mathcal{X}$ lying over a flat morphism $a : U \to U'$ of schemes. By Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-surjective-flat-locally-finite-presentation} there exists a jointly surjective family of \'etale morphisms $U'_i \to U'$ such that $U'_i \to U' \to \mathcal{X}$ factors through $\mathcal{X}_i$. Thus we obtain commutative diagrams $$ \xymatrix{ U_i = U \times_{U'} U_i' \ar[r]_-{a_i} \ar[d] & U_i' \ar[r]_{x_i'} \ar[d] & \mathcal{X}_i \ar[d]^{f_i} \\ U \ar[r]^a & U' \ar[r]^{x'} & \mathcal{X} } $$ Note that each $a_i$ is a flat morphism of schemes as a base change of $a$. Denote $\psi_i : x_i \to x'_i$ the morphism of $\mathcal{X}_i$ lying over $a_i$ with target $x_i'$. By assumption the comparison maps $c_{\psi_i} : (a_i)_{small}^*\big(f_i^*\mathcal{F}|_{(U'_i)_\etale}\big) \to f_i^*\mathcal{F}|_{(U_i)_\etale}$ is an isomorphism. Because the vertical arrows $U_i' \to U'$ and $U_i \to U$ are \'etale, the sheaves $f_i^*\mathcal{F}|_{(U_i')_\etale}$ and $f_i^*\mathcal{F}|_{(U_i)_\etale}$ are the restrictions of $\mathcal{F}|_{U'_\etale}$ and $\mathcal{F}|_{U_\etale}$ and the map $c_{\psi_i}$ is the restriction of $c_\varphi$ to $(U_i)_\etale$, see Sheaves on Stacks, Lemma \ref{stacks-sheaves-lemma-comparison}. Since $\{U_i \to U\}$ is an \'etale covering, this implies that the comparison map $c_\varphi$ is an isomorphism which is what we wanted to prove. \medskip\noindent Proof of (4). Let $\mathcal{I} \to \textit{Mod}(\mathcal{X}_\etale, \mathcal{O}_\mathcal{X})$, $i \mapsto \mathcal{F}_i$ be a diagram and assume each $\mathcal{F}_i$ has the flat base change property. Recall that $\colim_i \mathcal{F}_i$ is the sheafification of the presheaf colimit. As we are using the \'etale topology, it is clear that $$ (\colim_i \mathcal{F}_i)|_{U_\etale} = \colim_i {\mathcal{F}_i}|_{U_\etale} $$ As $f_{small}^*$ commutes with colimits (as a left adjoint) we see that (4) holds.