Joining a bulk superconductor with a spin-1/2 impurity realizes a microscopic model embodying the rich competition between magnetism and superconductivity, and leads to the formation of an isolated subgap bound state named the Yu-Shiba-Rusinov (YSR) state. The large tunability of semiconductor-superconductor hybrid devices, specifically InAs nanowires with epitaxial aluminium, allows for the creation of a double quantum dot geometry directly coupled to two superconducting leads, known as a S-DQD-S junction. When occupied by an odd number of electrons, each of these quantum dots host a single spin-1/2 degree of freedom enabling us to investigate the interaction between two YSR subgap states under tunable conditions. The electronic transport in the S-DQD-S junction combines aspects of single quasiparticle and cooperpair tunnelling, leading to detailed maps of differential conductance. Understanding these maps is not only a necessary requirement to characterize devices, but also reveals information about the non-equilibrium properties of the isolated quantum levels themselves. In this thesis we characterize and calculate transport in the S-DQD-S junction. This provides us with a window into the complicated dynamics of the two-impurity YSR state, and the various competitions at play. Using these tools we are able to classify charge diagrams of the double quantum dot by measurements of noise-dominated supercurrent, and by following the changes in these diagrams we realize the fully screened two-impurity YSR groundstate. Finite bias spectroscopy of YSR states is investigated using Keldysh Floquet Green’s functions capturing multiple Andreev reflections (MAR) to all orders. In this framework, we highlight the important roles of relaxation and poisoning in regard to subgap state transport, which are tested in measurements of direct transport between two opposing subgap states, where additional relaxation channels from MAR can be tuned on and off on demand.