Inspired by the microscopic control over dissipative processes in quantum optics and cold atoms, we develop an open-system framework to study dissipative control of transport in strongly interacting fermionic systems, relevant for both solid state and cold atom experiments. We show how subgap currents exhibiting Multiple Andreev Reflections -- the stimulated transport of electrons in the presence of Cooper-pairs -- can be controlled via engineering of superconducting leads or superfluid atomic gases. Our approach incorporates dissipation within the channel, which is naturally occurring and can be engineered in cold gas experiments. This opens opportunities for engineering many phenomena with transport in strongly interacting systems. As examples, we consider particle loss and dephasing, and note different behaviour for currents with different microscopic origin. We also show how to induce nonreciprocal electron and Cooper-pair currents.

Main text (5 pages, 4 figures) + Supplemental material (12 pages, 1 figure). Close to the published version