Choose a surjective smooth morphism $U \to \mathcal{X}$ for some scheme $U$. Then $U \to S$ is locally of finite presentation, see Morphisms of Stacks, Section \ref{stacks-morphisms-section-finite-presentation}. We can write $\mathcal{X} = [U/R]$ for some smooth groupoid in algebraic spaces $(U, R, s, t, c)$, see Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}. Since $U$ is locally of finite presentation over $S$ it follows that the algebraic space $R$ is locally of finite presentation over $S$. Recall that $[U/R]$ is the stack in groupoids over $(\Sch/S)_{fppf}$ obtained by stackyfying the category fibred in groupoids whose fibre category over $T$ is the groupoid $(U(T), R(T), s, t, c)$. Since $U$ and $R$ are limit preserving as functors (Limits of Spaces, Proposition \ref{spaces-limits-proposition-characterize-locally-finite-presentation}) this category fibred in groupoids is limit preserving. Thus it suffices to show that fppf stackyfication preserves the property of being limit preserving. This is true (hint: use Topologies, Lemma \ref{topologies-lemma-limit-fppf-topology}). However, we give a direct proof below using that in this case we know what the stackyfication amounts to. \medskip\noindent Let $T = \lim T_\lambda$ be a directed limit of affine schemes over $S$. We have to show that the functor $$ \colim [U/R]_{T_\lambda} \longrightarrow [U/R]_T $$ is an equivalence of categories. Let us show this functor is essentially surjective. Let $x \in \Ob([U/R]_T)$. In Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-stack-objects} the reader finds a description of the category $[U/R]_T$. In particular $x$ corresponds to an fppf covering $\{T_i \to T\}_{i \in I}$ and a $[U/R]$-descent datum $(u_i, r_{ij})$ relative to this covering. After refining this covering we may assume it is a standard fppf covering of the affine scheme $T$. By Topologies, Lemma \ref{topologies-lemma-limit-fppf-topology} we may choose a $\lambda$ and a standard fppf covering $\{T_{\lambda, i} \to T_\lambda\}_{i \in I}$ whose base change to $T$ is equal to $\{T_i \to T\}_{i \in I}$. For each $i$, after increasing $\lambda$, we can find a $u_{\lambda, i} : T_{\lambda, i} \to U$ whose composition with $T_i \to T_{\lambda, i}$ is the given morphism $u_i$ (this is where we use that $U$ is limit preserving). Similarly, for each $i, j$, after increasing $\lambda$, we can find a $r_{\lambda, ij} : T_{\lambda, i} \times_{T_\lambda} T_{\lambda, j} \to R$ whose composition with $T_{ij} \to T_{\lambda, ij}$ is the given morphism $r_{ij}$ (this is where we use that $R$ is limit preserving). After increasing $\lambda$ we can further assume that $$ s \circ r_{\lambda, ij} = u_{\lambda, i} \circ \text{pr}_0 \quad\text{and}\quad t \circ r_{\lambda, ij} = u_{\lambda, j} \circ \text{pr}_1, $$ and $$ c \circ (r_{\lambda, jk} \circ \text{pr}_{12}, r_{\lambda, ij} \circ \text{pr}_{01}) = r_{\lambda, ik} \circ \text{pr}_{02}. $$ In other words, we may assume that $(u_{\lambda, i}, r_{\lambda, ij})$ is a $[U/R]$-descent datum relative to the covering $\{T_{\lambda, i} \to T_\lambda\}_{i \in I}$. Then we obtain a corresponding object of $[U/R]$ over $T_\lambda$ whose pullback to $T$ is isomorphic to $x$ as desired. The proof of fully faithfulness works in exactly the same way using the description of morphisms in the fibre categories of $[U/T]$ given in Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-stack-objects}.