Using the mixed-spin system, we precisely calculate the dynamic critical phenomena, namely dynamic compensation behaviors, dynamic phase diagrams and dynamic hysteresis loops in a multilayer Ising system that composes of four monoatomic layers. As the method, we use the mean-field theory based on Glauber-type stochastic dynamics. It is found that the Hamiltonian parameters have a strong effect on the dynamic critical phenomena. The system displays the P-, Q-, R-, and S-type compensations and multiple hysteresis loop behaviors. The results are quali-tatively consistent with those of some theoretical and experimental studies.