A theoretical framework is developed for better understanding the time-dependent soft-X-ray response of dissipative quantum many-body systems. It is shown how X-ray absorption and resonant inelastic X-ray scattering (RIXS) at transition-metal Ledges can provide insight into ultrafast intersystem crossings of importance for energy conversion, ultrafast magnetism, and catalysis. The photoinduced doublet-to-quartet spin crossover on cobalt in Fe-Co Prussian blue analogues is used as an model system to demonstrate how the X-ray response is affected by the nonequilibrium dynamics on a femtosecond timescale. Changes in local spin and symmetry and the underlying mechanism are reflected in strong broadenings, a collapse of clear selection rules during the intersystem crossing, fluctuations in the isotropic branching ratio in X-ray absorption, crystal-field collapse/oscillations, and time-dependent anti-Stokes processes in RIXS.