A fully dynamic theoretical approach to layered optomagnonic structures, based on a time-Floquet scattering-matrix method, is developed. Its applicability is demonstrated on a simple design of a dual photonic-magnonic cavity, formed by sandwiching a magnetic garnet thin film between two dielectric Bragg mirrors, subject to continuous excitation of a perpendicular standing spin wave. Some remarkable phenomena, including nonlinear photon-magnon interaction effects and enhanced inelastic light scattering in the strong-coupling regime, fulfilling a triple-resonance condition, are analyzed and the limitations of the quasi-static adiabatic approximation are established.