The magnetic properties of a nonequilibrium mixed spin-2 and spin-5/2 Ising ferrimagnetic system with a crystal-field interaction (D) in the presence of a time-varying magnetic field on a hexagonal lattice are studied by using the Glauber-type stochastic dynamics. The model system consists of two interpenetrating sublattices with sigma=2 and S=5/2. The Hamiltonian model includes intersublattice, intrasublattice, and crystal-field interactions. The intersublattice interaction is considered antiferromagnetic to have a simple but interesting model of a ferrimagnetic system. The set of mean-field dynamic equations is obtained by employing the Glauber transition rates. First, we investigate the time variations in average sublattice magnetizations to find the phases in the system, and the temperature dependence of the dynamic sublattice magnetizations to characterize the nature (continuous or discontinuous) of the phase transitions and to obtain the dynamic phase transition points. Then, we study the temperature dependence of the total magnetization to find the dynamic compensation points as well as to determine the type of behavior. We also investigate the effect of a crystal-field interaction and the exchange couplings between the nearest-neighbor pairs of spins on the compensation phenomenon and present the dynamic phase diagrams. According to values of Hamiltonian parameters, the paramagnetic, the nonmagnetic, and the four different ferrimagnetic fundamental phases, seven different mixed phases, and the compensation temperature, or the N-type behavior in the Neacuteel classification nomenclature exist in the system. A comparison is made with the results of the available mixed spin Ising systems.