We simulate the evolution of relativistic eletrons injected into the medium of a small galaxy cluster by a central radio galaxy, studying how the initial jet power affects the dispersal and the emission properties of radio plasma. By coupling passive tracer particles to adaptive-mesh cosmological magnetohydrodynamic (MHD) simulations, we study how cosmic-ray electrons are dispersed as a function of the input jet power. We also investigate how the latter affects the thermal and non-thermal properties of the intracluster medium, with differences discernible up to about one Gyr after the start of the jet. We evolved the energy spectra of cosmic-ray electrons, subject to energy losses that are dominated by synchrotron and inverse Compton emission as well as energy gains via reacceleration by shock waves and turbulence. We find that in the absence of major mergers, the amount of re-acceleration experienced by cosmic-ray electrons is not enough to produce long-lived detectable radio emissions. However, for all simulations, the role of re-acceleration processes is crucial to maintaining a significant and volume-filling reservoir of fossil electrons (γ ∼ 10 3 ) for at least one Gyr after the first injection by jets. This is important in attempting to establish plausible explanations of recent discoveries of cluster-wide emission and other radio phenomena in galaxy clusters.