Functors between Kasparov categories from étale groupoid correspondences

For an étale correspondence $\Omega :G\to H$ of étale groupoids, we construct an induction functor ${\mathrm{Ind}}_{\Omega }:{\mathrm{KK}}^H\to {\mathrm{KK}}^G$ between equivariant Kasparov categories. We introduce the crossed product of an H-equivariant correspondence by Ω, and use this to build a natural transformation ${\alpha }_{\Omega }:K_{⁎}(G⋉{\mathrm{Ind}}_{\Omega }-)\Rightarrow K_{⁎}(H⋉-)$. When Ω is proper these constructions naturally sit above an induced map in K-theory $K_{⁎}\left(C^{⁎}\right(G\left)\right)\to K_{⁎}\left(C^{⁎}\right(H\left)\right)$.