Cascaded conical diffraction where optical elements modifying the local polarization state are intercalated between the aligned biaxial crystals is analyzed theoretically in the framework of paraxial diffraction theory. The obtained expressions are verified and confirmed experimentally for the case of a two-crystal cascade intercalated by a polarizer or a wave plate. The present approach can be used to realize a variety of vector beams with complex beam shapes composed of concentric rings with strongly modulated azimuthal intensity distribution. A potentially very fast switching of the overall beam shape is possible if the intercalated elements are electro-optically tunable retarders.