ABSTRACT Non-thermal emission from relativistic cosmic ray (CR) electrons gives insight into the strength and morphology of intra-cluster magnetic fields, as well as providing powerful tracers of structure formation shocks. Emission caused by CR protons on the other hand still challenges current observations and is therefore testing models of proton acceleration at intra-cluster shocks. Large-scale simulations including the effects of CRs have been difficult to achieve and have been mainly reduced to simulating an overall energy budget, or tracing CR populations in post-processing of simulation output and has often been done for either protons or electrons. We introduce crescendo: Cosmic Ray Evolution with SpeCtral Electrons aND prOtons, an efficient on-the-fly Fokker–Planck solver to evolve distributions of CR protons and electrons within every resolution element of our simulation. The solver accounts for CR (re-)acceleration at intra-cluster shocks, based on results of recent particle-in-cell simulations, adiabatic changes, and radiative losses of electrons. We show its performance in test cases as well as idealized galaxy cluster (GC) simulations. We apply the model to an idealized GC merger following best-fitting parameters for CIZA J2242.4 + 5301-1 and study CR injection, radio relic morphology, spectral steepening, and synchrotron emission.