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• Dynamic phase transitions of the mixed spin (1/2, 3/2) system are investigated by the path probability method. • The system undergoes two successive phase transitions in some cases. • The system exhibits one or two dynamic tricritical and double critical end points. • The system contains the paramagnetic, ferrimagnetic and mixed phases. • The system displays reentrant behavior for certain interaction parameter. Dynamic phase transitions (DPTs) of the mixed spin (1/2, 3/2) Ising ferrimagnetic model on the two interpenetrating square lattice with the bilinear and crystal-field interactions under a time-varying (sinusoidal) magnetic field were examined by using the path probability method (PPM). We studied stationary solutions of dynamic average magnetizations and obtained phases and then we investigated thermal behaviors of dynamic magnetizations to designate the nature of the DPTs and find the DPT temperatures. We also constructed the dynamic phase diagrams (DPDs) in two different planes. DPDs exhibit the paramagnetic (p), ferrimagnetic (i), and mixed phases (i + p), and one or two dynamic tricritical points and dynamic double critical end points. The reentrant behavior is also seen. We observed that the PPM is a more appropriate dynamic method to study the dynamic features of ferrimagnetism. [ABSTRACT FROM AUTHOR]