We formulate a scattering theory to study magnetic films in microwave cavities beyond the independent-spin and rotating-wave approximations of the Tavis-Cummings model. We demonstrate that strong coupling can be realized not only for the ferromagnetic resonance mode, but also for spin-wave resonances; the coupling strengths are mode dependent and decrease with increasing mode index. The strong-coupling regime can also be accessed electrically by spin pumping into a metal contact.