  We consider a free action of an Ore semigroup on a higher-rank graph, and the
induced action by endomorphisms of the $C^*$-algebra of the graph. We show that
the crossed product by this action is stably isomorphic to the $C^*$-algebra of
a quotient graph. Our main tool is Laca's dilation theory for endomorphic
actions of Ore semigroups on $C^*$-algebras, which embeds such an action in an
automorphic action of the enveloping group on a larger $C^*$-algebra.
