I am studying residual gerbes from Laumon Moret-Bailly and I would like to know if the residue field of the residue gerbe has the following property. I am a beginner in this subject so I find difficult to prove this assertion, although I think it should be true.

Let $\mathcal{X}$ be an algebraic stack of finite type over a field $k$, $\mathcal{G}$ be a residue gerbe of some point $x\in\mathcal{X}(L)$ ($L/k$ a field extension) with residue field $K$.

Is it true that there is a natural injection $K\to L$?

Is it true that $K$ is the intersection of all subfields of $L$ that are fields of definitions of objects that are isomorphic to $x$ on some extension $E/L$?

Also, is there a relation between the residue field and the field of moduli of $x$ (i.e., the intersection of all subfields of $L$ that are fields of definitions for $x$)?

If not, how can I get a handle on this residue field, in terms of the objects parametrized by $\mathcal{G}$?