Proof of (1). Choose $0 \in I$. As $X_0$ is quasi-compact, we can choose an affine scheme $W$ and an \'etale morphism $W \to B$ such that the image of $|X_0| \to |B|$ is contained in $|W| \to |B|$.  Choose an affine scheme $U_0$ and an \'etale morphism $U_0 \to X_0 \times_B W$ such that $U_0 \to X_0$ is surjective. (This is possible by our choice of $W$ and the fact that $X_0$ is quasi-compact; details omitted.) Let $V \to Y$, resp.\ $U \to X$, resp.\ $U_i \to X_i$ be the base change of $U_0 \to X_0$ (for $i \geq 0$). It suffices to prove that $V \to U_i$ is a closed immersion for $i$ sufficiently large. Thus we reduce to proving the result for $V \to U = \lim U_i$ over $W$. This follows from the case of schemes, which is Limits, Lemma \ref{limits-lemma-finite-type-eventually-closed}. \medskip\noindent Proof of (2). Choose $0 \in I$. Choose a quasi-compact open subspace $X'_0 \subset X_0$ such that $Y \to X_0$ factors through $X'_0$. After replacing $X_i$ by the inverse image of $X'_0$ for $i \geq 0$ we may assume all $X_i'$ are quasi-compact and quasi-separated. Let $U \subset X$ be a quasi-compact open such that $Y \to X$ factors through a closed immersion $Y \to U$ ($U$ exists as $Y$ is quasi-compact). By Lemma \ref{lemma-descend-opens} we may assume that $U = \lim U_i$ with $U_i \subset X_i$ quasi-compact open. By part (1) we see that $Y \to U_i$ is a closed immersion for some $i$. Thus (2) holds. \medskip\noindent Proof of (3). Choose $0 \in I$. Choose an affine scheme $U_0$ and a surjective \'etale morphism $U_0 \to X_0$. Set $U_i = X_i \times_{X_0} U_0$, $U = X \times_{X_0} U_0 = Y \times_{X_0} U_0$. Then $U = \lim U_i$ is a limit of affine schemes, the transition maps of the system are closed immersions, and $U \to U_0$ is of finite presentation (because $U \to B$ is locally of finite presentation and $U_0 \to B$ is locally of finite type and Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-finite-presentation-permanence}). Thus we've reduced to the following algebra fact: If $A = \lim A_i$ is a directed colimit of $R$-algebras with surjective transition maps and $A$ of finite presentation over $A_0$, then $A = A_i$ for some $i$. Namely, write $A = A_0/(f_1, \ldots, f_n)$. Pick $i$ such that $f_1, \ldots, f_n$ map to zero under the surjective map $A_0 \to A_i$. \medskip\noindent Proof of (4). Set $Z_i = Y \times_{X_i} Y$. As the transition morphisms $X_{i'} \to X_i$ are affine hence separated, the transition morphisms $Z_{i'} \to Z_i$ are closed immersions, see Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-fibre-product-after-map}. We have $\lim Z_i = Y \times_X Y = Y$ as $Y \to X$ is a monomorphism. Choose $0 \in I$. Since $Y \to X_0$ is locally of finite type (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-permanence-finite-type}) the morphism $Y \to Z_0$ is locally of finite presentation (Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-diagonal-morphism-finite-type}). The morphisms $Z_i \to Z_0$ are locally of finite type (they are closed immersions). Finally, $Z_i = Y \times_{X_i} Y$ is quasi-compact as $X_i$ is quasi-separated and $Y$ is quasi-compact. Thus part (3) applies to $Y = \lim_{i \geq 0} Z_i$ over $Z_0$ and we conclude $Y = Z_i$ for some $i$. This proves (4) and the lemma.