The propagation of excitonic polaritons is investigated in the nonlinear optical regime by real-time-resolved pump-and-probe measurements on a femtosecond time scale. High-quality ${\mathrm{ZnS}}_{\mathrm{x}}$${\mathrm{Se}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$/ZnSe/${\mathrm{ZnS}}_{\mathrm{x}}$${\mathrm{Se}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$ heterostructures are used which, in linear absorption, show clearly resolved Fabry-P\'erot resonances of polaritons confined in the ZnSe layer. Without pumping, the transmitted probe pulses exhibit a complex temporal beating behavior caused by quantum beating between heavy-hole and light-hole polaritons as well as by propagation beats. For increasing flux of a preceding pump pulse the transmitted power is temporally redistributed from later times toward the leading maximum of the pulse. Additionally, the amplitude of the beats continuously decreases. Almost quantitative agreement is found between the experimental data and calculations based on semiclassical polariton theory, clearly demonstrating the strong influence of spatial dispersion also on nonlinear polariton propagation. This allows determination of the density dependence of the oscillator strength, damping, and energy shift of the heavy-hole polariton.