In this paragraph we reduce to the case where $B$ is an affine scheme. Let $B' \to B$ be an \'etale morphism of algebraic spaces. Observe that conditions (1) and (2) are preserved if we replace $B$, $X_i$, $X$ by $B'$, $X_i \times_B B'$, $X \times_B B'$. Let $\{B_a \to B\}_{a \in A}$ be an \'etale covering with $B_a$ affine, see Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines}. For $a \in A$ denote $X_a$, $X_{a, i}$ the base changes of $X$ and the diagram to $B_a$. For $a, b \in A$ denote $X_{a, b}$ and $X_{a, b, i}$ the base changes of $X$ and the diagram to $B_a \times_B B_b$. By Lemma \ref{lemma-pushout-fpqc-local} it suffices to prove that $X_a = \colim X_{a, i}$ and $X_{a, b} = \colim X_{a, b, i}$. This reduces us to the case where $B = B_a$ (an affine scheme) or $B = B_a \times_B B_b$ (a separated scheme). Repeating the argument once more, we conclude that we may assume $B$ is an affine scheme (this uses that the intersection of affine opens in a separated scheme is affine). \medskip\noindent Assume $B$ is an affine scheme. Let $Z$ be an algebraic space over $B$. We have to show $$ \Mor_B(X, Z) \longrightarrow \lim \Mor_B(X_i, Z) $$ is a bijection. \medskip\noindent Proof of injectivity. Let $f, g : X \to Z$ be morphisms such that the compositions $f_i, g_i : X_i \to Z$ are the same for all $i$. Choose an affine scheme $Z'$ and an \'etale morphism $Z' \to Z$. By Properties of Spaces, Lemma \ref{spaces-properties-lemma-cover-by-union-affines} we know we can cover $Z$ by such affines. Set $U = X \times_{f, Z} Z'$ and $U' = X \times_{g, Z} Z'$ and denote $p : U \to X$ and $p' : U' \to X$ the projections. Since $f_i = g_i$ for all $i$, we see that $$ U_i = X_i \times_{f_i, Z} Z' = X_i \times_{g_i, Z} Z' = U'_i $$ compatible with transition morphisms. By (1) there is a unique isomorphism $\epsilon : U \to U'$ as algebraic spaces over $X$, i.e., with $p = p' \circ \epsilon$ which is compatible with the displayed identifications. Choose an \'etale covering $\{h_a : U_a \to U\}$ with $U_a$ affine. By (2) we see that $f \circ p \circ h_a = g \circ p' \circ \epsilon \circ h_a = g \circ p \circ h_a$. Since $\{h_a : U_a \to U\}$ is an \'etale covering we conclude $f \circ p = g \circ p$. Since the collection of morphisms $p : U \to X$ we obtain in this manner is an \'etale covering, we conclude that $f = g$. \medskip\noindent Proof of surjectivity. Let $f_i : X_i \to Z$ be an element of the right hand side of the displayed arrow in the first paragraph of the proof. It suffices to find an \'etale covering $\{U_c \to X\}_{c \in C}$ such that the families $f_{c, i} \in \lim_i \Mor_B(X_i \times_X U_c, Z)$ come from morphisms $f_c : U_c \to Z$. Namely, by the uniqueness proved above the morphisms $f_c$ will agree on $U_c \times_X U_b$ and hence will descend to give the desired morphism $f : X \to Z$. To find our covering, we first choose an \'etale covering $\{g_a : Z_a \to Z\}_{a \in A}$ where each $Z_a$ is affine. Then we let $U_{a, i} = X_i \times_{f_i, Z} Z_a$. By (1) we find $U_{a, i} = X_i \times_X U_a$ for some algebraic spaces $U_a$ \'etale over $X$. Then we choose \'etale coverings $\{U_{a, b} \to U_a\}_{b \in B_a}$ with $U_{a, b}$ affine and we consider the morphisms $$ U_{a, b, i} = X_i \times_X U_{a, b} \to X_i \times_X U_a = X_i \times_{f_i, Z} Z_a \to Z_a $$ By (2) we obtain morphisms $f_{a, b} : U_{a, b} \to Z_a$ compatible with these morphisms. Setting $C = \coprod_{a \in A} B_a$ and for $c \in C$ corresponding to $b \in B_a$ setting $U_c = U_{a, b}$ and $f_c = g_a \circ f_{a, b} : U_c \to Z$ we conclude.