We characterize when there exists a diagonal-preserving (Formula presented.)-isomorphism between two graph (Formula presented.)-algebras in terms of the dynamics of the boundary path spaces. In particular, we refine the notion of ‘orbit equivalence’ between the boundary path spaces of the directed graphs (Formula presented.) and (Formula presented.) and show that this is a necessary and sufficient condition for the existence of a diagonal-preserving (Formula presented.)-isomorphism between the graph (Formula presented.)-algebras (Formula presented.) and (Formula presented.).