We show that hybrid structures of topological insulators (TI) and materials without topological protection can be employed to create perfectly conducting channels (PCCs) hosted in the non-topological part. These states inherit the topological protection from the proximity of the TI but are more fragile to time-reversal symmetry breaking because of their extended character. We explore their formation in the band structure of model hybrid systems as well as realistic heterostructures involving HgTe/CdTe-based two-dimensional TIs and show how their appearance can be understood in terms of an effective boundary condition. Using numerical quantum transport calculations for the HgTe/CdTe material system we propose two experimental settings which allow for the detection of the induced PCCs, both in the localized and diffusive regime, by means of magneto conductance and shot noise.