© 2022, The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.The multilayer Ising model which is one of the most important research topics in both statistical mechanics and solid-state physics is used in the study of surface and finite size effects. Although there have been many studies on the equilibrium magnetic properties of this phenomenon due to a large number of technological application potential, the studies on the out-of-equilibrium magnetic properties are limited. Hence, the mean-field theory based on Glauber-type stochastic dynamics is used to investigate the out-of-equilibrium magnetic properties of the multilayer mixed spin-1 and spin-3/2 Ising model. The system is a four-monatomic-layered square structure in which two kinds of spins (spin-1 and spin-3/2) occupy sites alternately. For this purpose, we examine the thermal behavior of dynamic magnetizations to obtain dynamic phase transitions (DPTs) and find the type of DPTs. The dynamic total magnetization as a function of temperature is studied to determine the dynamic compensation types of multilayer mixed spin-1 and spin-3/2 Ising model and found the Q-, P-, R-, S- and N-type compensations behaviors. Finally, we calculate the dynamic phase diagrams (DPDs) of the multilayer mixed Ising model for five various planes and obtain that the DPDs include five fundamental phases, nine mixed phases (or regions), and special points. We see that the DPTs and DPDs are strongly dependent on Hamiltonian parameters.