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 ⋉ X ⋊ H. Given a separable groupoid dynamical system (A, G ⋉ X ⋊ H, ω) we may restrict ω to an action of G ⋉ X on A and form the crossed product A ⋊ G ⋉ X. We show that there is an action of H on A ⋊ G ⋉ X and that the iterated crossed product (A ⋊ G ⋉ X) ⋊H is naturally isomorphic to the crossed product A ⋊ (G ⋉ X ⋊ H).