Explicit construction of orbit equivalence for rank-one actions

Let $G$ be a discrete countable infinite group. Given two rank-one measure-preserving $G$-actions whose invariant measures are either both finite or both infinite, we give a direct explicit construction, from their cutting-and-stacking parameters, of a Borel orbit equivalence on invariant conull subsets. We also prove a topological counterpart of this assertion, under additional compatibility assumptions, for continuous $(C,F)$-actions of $G$ on non-compact locally compact Cantor spaces.