Write $Y = \lim_{i \in I} Y_i$ as a limit of algebraic spaces over a directed set $I$ with affine transition morphisms and with $Y_i$ Noetherian, see Limits of Spaces, Proposition \ref{spaces-limits-proposition-approximate}. We will use the material from Limits of Spaces, Section \ref{spaces-limits-section-finite-type-quasi-separated}. \medskip\noindent Choose a presentation $\mathcal{X} = [U/R]$. Denote $(U, R, s, t, c, e, i)$ the corresponding groupoid in algebraic spaces over $Y$. We may and do assume $U$ is affine. Then $U$, $R$, $R \times_{s, U, t} R$ are quasi-separated algebraic spaces of finite type over $Y$. We have two morpisms $s, t : R \to U$, three morphisms $c : R \times_{s, U, t} R \to R$, $\text{pr}_1 : R \times_{s, U, t} R \to R$, $\text{pr}_2 : R \times_{s, U, t} R \to R$, a morphism $e : U \to R$, and finally a morphism $i : R \to R$. These morphisms satisfy a list of axioms which are detailed in Groupoids, Section \ref{groupoids-section-groupoids}. \medskip\noindent According to Limits of Spaces, Remark \ref{spaces-limits-remark-finite-type-gives-well-defined-system} we can find an $i_0 \in I$ and inverse systems \begin{enumerate} \item $(U_i)_{i \geq i_0}$, \item $(R_i)_{i \geq i_0}$, \item $(T_i)_{i \geq i_0}$ \end{enumerate} over $(Y_i)_{i \geq i_0}$ such that $U = \lim_{i \geq i_0} U_i$, $R = \lim_{i \geq i_0} R_i$, and $R \times_{s, U, t} R = \lim_{i \geq i_0} T_i$ and such that there exist morphisms of systems \begin{enumerate} \item $(s_i)_{i \geq i_0} : (R_i)_{i \geq i_0} \to (U_i)_{i \geq i_0}$, \item $(t_i)_{i \geq i_0} : (R_i)_{i \geq i_0} \to (U_i)_{i \geq i_0}$, \item $(c_i)_{i \geq i_0} : (T_i)_{i \geq i_0} \to (R_i)_{i \geq i_0}$, \item $(p_i)_{i \geq i_0} : (T_i)_{i \geq i_0} \to (R_i)_{i \geq i_0}$, \item $(q_i)_{i \geq i_0} : (T_i)_{i \geq i_0} \to (R_i)_{i \geq i_0}$, \item $(e_i)_{i \geq i_0} : (U_i)_{i \geq i_0} \to (R_i)_{i \geq i_0}$, \item $(i_i)_{i \geq i_0} : (R_i)_{i \geq i_0} \to (R_i)_{i \geq i_0}$ \end{enumerate} with $s = \lim_{i \geq i_0} s_i$, $t = \lim_{i \geq i_0} t_i$, $c = \lim_{i \geq i_0} c_i$, $\text{pr}_1 = \lim_{i \geq i_0} p_i$, $\text{pr}_2 = \lim_{i \geq i_0} q_i$, $e = \lim_{i \geq i_0} e_i$, and $i = \lim_{i \geq i_0} i_i$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-morphism-good-diagram-smooth} we see that we may assume that $s_i$ and $t_i$ are smooth (this may require increasing $i_0$). By Limits of Spaces, Lemma \ref{spaces-limits-lemma-morphism-good-diagram-flat} we may assume that the maps $R \to U \times_{U_i, s_i} R_i$ given by $s$ and $R \to R_i$ and $R \to U \times_{U_i, t_i} R_i$ given by $t$ and $R \to R_i$ are isomorphisms for all $i \geq i_0$. By Limits of Spaces, Lemma \ref{spaces-limits-lemma-good-diagram-fibre-product} we see that we may assume that the diagrams $$ \xymatrix{ T_i \ar[r]_{q_i} \ar[d]_{p_i} & R_i \ar[d]^{t_i} \\ R_i \ar[r]^{s_i} & U_i } $$ are cartesian. The uniqueness of Limits of Spaces, Lemma \ref{spaces-limits-lemma-morphism-good-diagram} then guarantees that for a sufficiently large $i$ the relations between the morphisms $s, t, c, e, i$ mentioned above are satisfied by $s_i, t_i, c_i, e_i, i_i$. Fix such an $i$. \medskip\noindent It follows that $(U_i, R_i, s_i, t_i, c_i, e_i, i_i)$ is a smooth groupoid in algebraic spaces over $Y_i$. Hence $\mathcal{X}_i = [U_i/R_i]$ is an algebraic stack (Algebraic Stacks, Theorem \ref{algebraic-theorem-smooth-groupoid-gives-algebraic-stack}). The morphism of groupoids $$ (U, R, s, t, c, e, i) \to (U_i, R_i, s_i, t_i, c_i, e_i, i_i) $$ over $Y \to Y_i$ determines a commutative diagram $$ \xymatrix{ \mathcal{X} \ar[d] \ar[r] & \mathcal{X}_i \ar[d] \\ Y \ar[r] & Y_i } $$ (Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-stack-functorial}). We claim that the morphism $\mathcal{X} \to Y \times_{Y_i} \mathcal{X}_i$ is a closed immersion. The claim finishes the proof because the algebraic stack $\mathcal{X}_i \to Y_i$ is of finite presentation by construction. To prove the claim, note that the left diagram $$ \xymatrix{ U \ar[d] \ar[r] & U_i \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X}_i } \quad\quad \xymatrix{ U \ar[d] \ar[r] & Y \times_{Y_i} U_i \ar[d] \\ \mathcal{X} \ar[r] & Y \times_{Y_i} \mathcal{X}_i } $$ is cartesian by Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-criterion-fibre-product} and the results mentioned above. Hence the right commutative diagram is cartesian too. Then the desired result follows from the fact that $U \to Y \times_{Y_i} U_i$ is a closed immersion by construction of the inverse system $(U_i)$ in Limits of Spaces, Lemma \ref{spaces-limits-lemma-limit-from-good-diagram}, the fact that $Y \times_{Y_i} U_i \to Y \times_{Y_i} \mathcal{X}_i$ is smooth and surjective, and Properties of Stacks, Lemma \ref{stacks-properties-lemma-check-immersion-covering}.