  This paper investigates the $\mathrm{K}$-theory of twisted groupoid
$\mathrm{C}^*$-algebras. It is shown that a homotopy of twists on an ample
groupoid satisfying the Baum-Connes conjecture with coefficients gives rise to
an isomorphism between the $\mathrm{K}$-theory groups of the respective twisted
groupoid $\mathrm{C}^*$-algebras. The results are also interpreted in an
inverse semigroup setting and applied to generalized Renault-Deaconu groupoids
and $P$-graph algebras.
