  We study the notions of continuous orbit equivalence and eventual one-sided
conjugacy of finitely-aligned higher-rank graphs and two-sided conjugacy of
row-finite higher-rank graphs with finitely many vertices and no sinks or
sources. We show that there is a continuous orbit equivalence between two
finitely-aligned higher-rank graphs that preserves the periodicity of boundary
paths if and only if the boundary path groupoids are isomorphic, and
characterise continuous orbit equivalence, eventual one-sided conjugacy, and
two-sided conjugacy of higher-rank graphs in terms of the $C^*$-algebras and
the Kumjian-Pask algebras of the higher-rank graphs.
