Temperature dependent magnetic properties of superconductor-ferromagnet-superconductor (SC/FM/SC) trilayers are studied both experimentally and theoretically, with a focus on ferromagnetic resonance (FMR). The influence of the SC and FM layer thicknesses on the FMR field is examined. To differentiate the mechanisms involved, we additionally investigate structures containing nonmagnetic metallic (M) or insulating (I) spacers (SC/FM/M/SC or SC/FM/I/SC). All the studied multilayers show large reductions in the FMR field below the critical temperature of the SC, except the system containing an insulating spacer (SC/FM/I/SC). This SC-induced FMR-shift (resonance field/frequency) is larger for thicker SC as well as FM layers, reaching a saturation value for very large thicknesses. To explain the measured results, an analytical model is developed, in which the FM-magnetization precession modulates the magnetic flux in the system, thereby inducing an alternating supercurrent in the SC, which in turn produces a dynamic back-action magnetic field on the FM that shifts its resonance frequency. The model considers closed current loops, where the FM layer conductively links the supercurrents flowing in the opposite directions in the two outer SC layers. Our results provide a practical route for increasing the operating frequency of magnonic devices.