  We prove that for torsion-free amenable ample groupoids, an isomorphism in
groupoid homology induced by an \'etale correspondence yields an isomorphism in
the K-theory of the associated $\mathrm{C}^\ast$-algebras. We apply this to
extend X. Li's K-theory formula for left regular inverse semigroup
$\mathrm{C}^\ast$-algebras. These results are obtained by developing the
functoriality of the ABC spectral sequence.
