  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.
