Functors between Kasparov categories from \'etale groupoid correspondences

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