Let $G$ be a locally compact Hausdorff (second countable) groupoid with Hausdorff (second countable) unit space $X$. Assume $G$ is étale, i.e., the source and range maps of $G$ are local homeomorphisms. We say that $G$ is proper if the map $(s,r)\colon G \to X\times X$ is proper, i.e., the preimage of a compact subset is compact.
I think the following theorem is true, but I can't find a proof. I'd greatly appreciate any pointer.
if $G$ is proper then each $x\in X$ admits an open neighborhood $U\subseteq X$ equipped with an action of $G_x^x$ (the automorphisms group at $x$) such that the restriction $G|_U$ is isomorphic to the action groupoid $U\rtimes G_x^x$ and the map (whenever it is defined) $$ G\times_{G_x^x}U \to X$$ sending $[g,x]$ to $gx$ is a $G$-equivariant homeomorphism onto an open neighborhood of (the orbit of) $x\in X$.
EDIT: I would be interested to know if anything is known in the non-Lie non-étale case. When $G$ is a locally compact group (acting properly), a result of Abels says that $X$ is locally compactly induced, but the compact subgroup cannot (in general) be chosen to be a stabilizer. By analogy, for a locally compact proper groupoid $G$ it seems reasonable to prove that
$G$ is locally Morita equivalent to an action groupoid by a compact group. More precisely, $X$ can be covered by opens $V$ which such that $G|_V$ is Morita equivalent to $U\rtimes H$ for $H$ a compact group acting on a subset $U\subseteq V$. Note that $U$ (the slice) need not be open.
