Electronic excitations can be efficiently analyzed in terms of the underlying Kohn-Sham (KS) electron-hole transitions. While such a decomposition is readily available in the linear-response time-dependent density-functional theory (TDDFT) approaches based on the Casida equations, a comparable analysis is less commonly conducted within the real-time-propagation TDDFT (RT-TDDFT). To improve this situation, we present here an implementation of a KS decomposition tool within the local-basis-set RT-TDDFT code in the free GPAW package. Our implementation is based on postprocessing of data that is readily available during time propagation, which is important for retaining the efficiency of the underlying RT-TDDFT to large systems. After benchmarking our implementation on small benzene derivatives by explicitly reconstructing the Casida eigenvectors from RT-TDDFT, we demonstrate the performance of the method by analyzing the plasmon resonances of icosahedral silver nanoparticles up to Ag561. The method provides a clear description of the splitting of the plasmon in small nanoparticles due to individual single-electron transitions as well as the formation of a distinct d-electron-screened plasmon resonance in larger nanoparticles.