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

Skew-products of higher-rank graphs and crossed products by semigroups