Functors between Kasparov categories from étale groupoid correspondences

For an étale correspondence $\mathrm{\Omega }\text{\hspace{0.17em}⁣}:G\to H$ of étale groupoids, we construct an induction functor ${\mathrm{I}\mathrm{n}\mathrm{d}}_{\mathrm{\Omega }}\text{\hspace{0.17em}⁣}:{\mathrm{K}\mathrm{K}}^H\to {\mathrm{K}\mathrm{K}}^G$ between equivariant Kasparov categories. We introduce the crossed product of an $H$-equivariant correspondence by $\mathrm{\Omega }$, and use this to build a natural transformation ${\alpha }_{\mathrm{\Omega }}\text{\hspace{0.17em}⁣}:K_{\ast }(G⋉{\mathrm{I}\mathrm{n}\mathrm{d}}_{\mathrm{\Omega }}-)\Rightarrow K_{\ast }(H⋉-)$. When $\mathrm{\Omega }$ is proper these constructions naturally sit above an induced map in K-theory $K_{\ast }(C^{\ast }(G))\to K_{\ast }(C^{\ast }(H))$.