Properties that descend along smooth morphisms Let $U\to \mathcal X$ be a smooth surjective (representable) morphism from a scheme to an algebraic stack. Let $f:\mathcal Y\to \mathcal X$ be any representable morphism of algebraic stacks. The induced map $$g:\mathcal Y\times_\mathcal XU\to U$$ is a morphism of schemes.

Question. Is it true that if $g$ is étale then $f$ was étale? In
  general, what kind of properties of $g$ will descend to $f$?

Thanks!
 A: Let $\mathcal P$ be a property of morphisms that is local for the smooth topology; that is, $\mathcal P$ is stable under smooth base change, and given a morphism
$$f \colon X \to Y$$
of schemes and a smooth cover $\{U_i \to Y\}_{i \in I}$, if each pullback $X \times_Y U_i \to U_i$ satisfies $\mathcal P$, then so does $f$.
Lemma. Let $f \colon \mathcal Y \to \mathcal X$ be a representable morphism of stacks. Let $U \to \mathcal X$ be a smooth surjective (representable) morphism from a scheme $U$. Let $\mathcal P$ be a property that is local for the smooth topology. Then $f$ has property $\mathcal P$ if and only if the base change $g \colon U \times_{\mathcal Y} \mathcal X \to U$ does.
Proof. One implication is obvious, so we will assume that $g$ has property $\mathcal P$. 
Let $V \to \mathcal X$ be any morphism from a scheme to $\mathcal X$, and let $h \colon V \times_{\mathcal X} \mathcal Y \to V$ be the corresponding morphism of schemes. We need to show that $h$ has property $\mathcal P$.
To do this, we may work locally on $V$. In particular, we use the surjective smooth morphism $U \to \mathcal X$ and base change to $V$ to get a surjective smooth morphism of schemes $U \times_{\mathcal X} V \to V$. This will be our smooth cover. Consider the pullback diagram
$$\begin{array}{ccc}U \times_{\mathcal X} V \times_{\mathcal X} \mathcal Y & \to & U \times_{\mathcal X} V\\ \downarrow & & \downarrow \\V \times_{\mathcal X} \mathcal Y & \stackrel h \to &\  V.\end{array}$$
The top map is the pullback of $g \colon U \times_{\mathcal X} \mathcal Y \to U$ along $U \times_{\mathcal X} V \to U$, hence has property $\mathcal P$ since $g$ does. Since $\mathcal P$ is a local property for the smooth topology, it follows that $h$ has property $\mathcal P$. $\square$
Remark. This is exactly the same proof as you always use in the Zariski topology for checking that properties like locally of finite presentation, affine, finite, etc. can be checked on any one open cover. The Stacks project has a systematic exposition of it; see here for morphisms of schemes for the Zariski topology, and here for the fpqc topology on schemes.
I guess I just proved (and this is probably in the Stacks project as well) that such properties are automatically local on representable morphisms of stacks.
Remark. A good place to find many examples of properties $\mathcal P$ that are even fpqc-local on the target is EGA IV$_2$, Prop. 2.7.1. Weirdly, étale is not listed here (nor are unramified and flat), but it's not so hard to prove these yourself (or you could look in the Stacks projects).
