  For an $r$-discrete Hausdorff groupoid ${\cal G}$ and an inverse semigroup
$S$ of slices of ${\cal G}$ there is an isomorphism between ${\cal
G}$-equivariant $KK$-theory and compatible $S$-equivariant $KK$-theory. We use
it to define descent homomorphisms for $S$, and indicate a Baum--Connes map for
inverse semigroups. Also findings by Khoshkam and Skandalis for crossed
products by inverse semigroups are reflected in $KK$-theory.
