My question is about the passage (11.1) in the book of Laumon and Moret-Bailly on algebraic stacks. There we have a scheme $S$, an algebraic stack $\mathscr{X}$ over $S$, and a point $\xi$ of $\mathscr{X}$. The residual gerbe $\mathscr{G}_\xi$ at $\xi$ is defined as follows: choose an $S$-field $K$ and an $S$-morphism $x\colon \mathrm{Spec}(K) \rightarrow \mathscr{X}$ representing $\xi$, and let $\mathscr{G}_\xi \subset \mathscr{X}$ be the smallest substack (for the fppf topology) through which $x$ factors. The claim is that this $\mathscr{G}_\xi$ does not depend on the choices of $K$ and $x$. How does one verify this?

I am especially having trouble in the case when $K'$ is a big overfield of $K$, when I cannot figure out how to make contact with the fppf topology that is used in defining $\mathscr{G}_\xi$. It seems to me that $\mathscr{G}_\xi$ could shrink after replacing $K$ by this $K'$ in the defining procedure.