We prove that the boundary extension of a quasi-isometry of hyperbolic space which conjugates two Kleinian groups maps the Myrberg limit set of the first group bijectively onto the Myrberg limit set of the second group. As a first application of this observation we obtain new geometric proofs of Mostow's Rigidity Theorem. Furthermore, we recover the classical result that motions in the Teichmüller space of a compact hyperbolic surface induce singular boundary maps.