Residual gerbe and field of moduli

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 $\mathcal{G}$?

(1): There is a natural injection $K\to L$, indeed we have $\operatorname{Spec} L\to \mathcal{G}\to \operatorname{Spec} K$.
(2) and (3): No, it can happen that there are no objects defined on any strict subfield of $L$. For example, consider the $\mu_2$-gerbe $\mathcal{G}\to\mathbb{R}$ corresponding to the Brauer class of the usual quaternions (explicitly: $\mathcal{G}=[\mathbb{C}/\mu_4]$ where $\operatorname{deg}(i)=2$). Take $L=\mathbb{C}$. Then we have an $L$-point but no $K=\mathbb{R}$-point.
You can obtain $\operatorname{Spec} K$ as the coequalizer of $\operatorname{Spec} L\times_\mathcal{G} \operatorname{Spec} L\rightrightarrows \operatorname{Spec} L$ in the category of schemes (that is $K$ is the equalizer of $L\rightrightarrows \Gamma(\operatorname{Spec} L\times_\mathcal{G} \operatorname{Spec} L)$). Think of the case where $\mathcal{G}=\operatorname{Spec} K$, then $\operatorname{Spec} L\times_\mathcal{G} \operatorname{Spec} L$ is a "Galois groupoid" (the usual action groupoid of $\operatorname{Gal}(L/K)$ if $L/K$ is Galois). More precisely, the stabilizer group acts freely on $\operatorname{Spec} L\times_\mathcal{G} \operatorname{Spec} L$ and the quotient is the stabilizer groupoid of $L/K$.
In the example above, the groupoid becomes the action of $\mu_4$ on $\mathbb{C}$ and $\mu_4$ is the extension of the Galois group $\operatorname{Gal}(L/K)$ and the stabilizer group $\mu_2$.