We demonstrate direct transport between two opposing sets of Yu-Shiba-Rusinov (YSR) subgap states realized in a double quantum dot. This bound-state-to-bound-state transport relies on intrinsic quasiparticle relaxation, and the tunable gating of this quantum dot device allows us to explore also an additional relaxation mechanism based on charge transferring Andreev reflections. The transition between these two relaxation regimes is identified in the experiment as a marked gate-induced stepwise change in conductance. We present a transport calculation, including YSR bound states and multiple Andreev reflections alongside quasiparticle relaxation, due to a weak tunnel coupling to a nearby normal metal, and obtain excellent agreement with the data.