We study the nonequilibrium aspects of the kinetics of a mixed spin-1 and spin-2 Ising system including the biquadratic nearest-neighbor pair interaction (K) and crystal-field interaction (D) under the presence of a time-dependent oscillating external magnetic field within a mean-field approach. In particular, we calculate the dynamic phase transition (DPT) temperatures and present dynamic phase diagrams. The set of mean-field dynamic equations is obtained by employing Glauber transition rates. We investigate the time variations of average order parameters to find the phases in the system. We also study the thermal behavior of the dynamic order parameters to characterize the nature (continuous or discontinuous) of the phase transitions and obtain the DPT points. The dynamic phase diagrams are presented in three different planes. Phase diagrams contain disordered (d), ferrimagnetic (i), antiquadrupolar or staggered (a) phases, and four coexistence or mixed phase regions, namely i + d, i + a, i + a + d and a + d mixed phases, that strongly depend on interaction parameters.