The alteration of the dielectric membrane properties by membrane engineering techniques such as carbon nanotube (CNT) coating opens the way to novel molecular transport strategies for biosensing purposes. In this article, we predict a macromolecular transport mechanism enabling the dielectric manipulation of the polymer translocation dynamics in dielectric membrane pores confining mixed electrolytes. In the giant permittivity regime of these engineered membranes governed by attractive polarization forces, multivalent ions adsorbed by the membrane nanopore trigger a monovalent ion separation and set an electroosmotic counterion flow. The drag force exerted by this flow is sufficiently strong to suppress and invert the electrophoretic velocity of anionic polymers and also to generate the mobility of neutral polymers whose speed and direction can be solely adjusted by the charge and concentration of the added multivalent ions. These features identify the dielectrically generated transport mechanism as an efficient means to drive overall neutral or weakly charged analytes that cannot be controlled by an external voltage. We also reveal that, in anionic polymer translocation, multivalent cation addition into the monovalent salt solution amplifies the electric current signal by several factors. The signal amplification is caused by the electrostatic many-body interactions replacing the monovalent polymer counterions by the multivalent cations of higher electric mobility. The strength of this electrokinetic charge discrimination points out the potential of multivalent ions as current amplifiers capable of providing boosted resolution in nanopore-based biosensing techniques.