The nonequilibrium behavior of the mixed spin-3/2 and spin-2 Ising system on the bilayer square lattice under a time-varying magnetic field is studied by using the Glauber-type stochastic dynamics. The dynamic equations describing the behavior of the system are derived by utilizing the Master equation and Glauber transition rates. The time variations of average magnetizations and the thermal variations of the dynamic magnetizations are investigated to obtain the dynamic phase diagrams. The dynamic phase diagrams are constructed in four different planes for the ferromagnetic/ferromagnetic (FM/FM), antiferromagnetic/ferromagnetic (AFM/FM) and antiferromagnetic/antiferromagnetic (AFM/AFM) interactions and the effects of the Hamiltonian parameters on the dynamic critical behavior of the system are studied. It is observed that the system exhibits seven fundamental phases and twenty five mixed phases which are composed of binary, ternary and tetrad combinations of fundamental phases. It is also found that the dynamic phase diagrams contain both first- and second-order phase transitions besides dynamic tricritical point, triple point (TP), quadruple point (QP), double critical end point (B), zero temperature critical point (Z), multicritical point (A) and tetracritical point (M). The reentrant behavior occurs for the FM/FM interaction. (C) 2014 Elsevier B.V. All rights reserved.