For an étale correspondence Ω:G→H of étale groupoids, we construct an induction functor IndΩ:KKH→KKG between equivariant Kasparov categories. We introduce the crossed product of an H-equivariant correspondence by Ω, and use this to build a natural transformation αΩ:K⁎(G⋉IndΩ−)⇒K⁎(H⋉−). When Ω is proper these constructions naturally sit above an induced map in K-theory K⁎(C⁎(G))→K⁎(C⁎(H)).