Groupoid equivalence and the associated iterated crossed product
  Given groupoids $G$ and $H$ and a $(G,H)$-equivalence $X$ we may form the transformation groupoid $G\ltimes X\rtimes H$. Given a separable groupoid dynamical system $(A,G\ltimes X\rtimes H,\omega)$ we may restrict $\omega$ to an action of $G\ltimes X$ on $A$ and form the crossed product $A\rtimes G\ltimes X$. We show that there is an action of $H$ on $A\rtimes G\ltimes X$ and that the iterated crossed product $(A\rtimes G\ltimes X)\rtimes H$ is naturally isomorphic to the crossed product $A\rtimes (G\ltimes X\rtimes H)$. 
Jonathan Henry Brown, Geoff Goehle and Dana P. Williams
math.OA