  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 $K_*(C^*(G))
\to K_*(C^*(H))$.
