An étale morphism of schemes is unramified, and an unramified morphism is locally quasi-finite. Does the same hold for étale morphisms of algebraic stacks?
Let us recall the definitions, following the Stacks Project[1].
Suppose $P$ is a property of morphisms of schemes. For some common Grothendieck topologies $\tau$ on the category of schemes, there are notions of $P$ being $\tau$-local on the source, and $\tau$-local on the target/base. Let's call a property $P$ of morphisms $\tau$-local, if it is $\tau$-local on the source, $\tau$-local on the base, and stable under post-composition by open immersions. The Stacks Project calls this "$\tau$-local on the source and target", which seems to conflict with ordinary language.
The property of being locally quasi-finite is étale-local in the above sense, so it can be defined for an arbitrary morphism of algebraic spaces. It is fpqc-local on the base and stable under base change, so it can also be defined for morphisms of algebraic stacks that are representable by algebraic spaces. But it's not smooth-local, which means the definition doesn't bootstrap automatically to arbitrary morphisms of algebraic stacks: it has to be given a separate definition. Here's how it's defined in the Stacks Project.
Definition: A morphism $f: \cal{X} \rightarrow \cal{Y}$ of algebraic stacks is locally quasi-finite if it is locally of finite type, quasi-DM, and for every morphism $\rm{Spec}(k) \rightarrow \cal{Y}$, where $k$ is a field, the space $|\cal{X}_k|$ is discrete.
Here quasi-DM means the diagonal $\Delta_f$ is locally quasi-finite (as a morphism representable by algebraic spaces), and $|\cal{X}_k|$ is the "underlying topological space" of the fibre $\cal{X}_k$, consisting of equivalence classes of morphisms from the spectra of fields.
Question: With the above definition, is an étale morphism of algebraic stacks locally quasi-finite?
An acceptable answer would either give a clear proof why the statement is true, provide an enlightening counter-example why it's not, or point to a citable source where the problem is resolved. I have looked in LMB [2], but there locally quasi-finite is only mentioned in the appendix for algebraic spaces, not for stacks.
In fact for my purpose, $\cal{X}$ and $\cal{Y}$ are Deligne-Mumford stacks, and in that case the question may be easier. Since locally quasi-finite is an étale-local property, there should be another way to define it for morphisms of Deligne-Mumford stacks by passing to étale presentations. Then there's probably some general rule transferring implication relations between étale-local properties of morphisms from schemes to Deligne-Mumford stacks, which would give (étale $\Rightarrow$ locally quasi-finite). I assume this is not done in the Stacks Project because there the emphasis is on general algebraic stacks. If all that is indeed the case, the question becomes: is that (presumably easier) definition of locally quasi-finite for Deligne-Mumford stacks compatible with the one above for general algebraic stacks?
[1]: Stacks Project (http://stacks.math.columbia.edu/)
[2]: Champes algébriques, Gérard Laumon, Laurent Moret-Bailley
EDIT: The bounty is just to draw attention. This question is probably not hard for experts in algebraic stacks.
Yes. Let $V \to \mathcal{Y}$ be a morphism from an algebraic space. Then the base change $\mathcal{X}_V$ admits an étale covering $U \to \mathcal{X}_V$ by an algebraic space $U$ such that the composition $U \to V$ is étale. In particular, $\mathcal{X}_V$ is a Deligne-Mumford stack, so $f$ is DM and a fortiori quasi-DM. Furthermore, $U \to V$ is quasi-finite in the usual sense, so $f$ is quasi-finite by [SP TAG 06UF].
Note that the descent type definition makes sense for any DM morphism (in particular for any morphism of Deligne-Mumford stacks). The argument above shows that the defintion using descent is equivalent to the more general definition in this situation.