Three-dimensional (3D) strongly correlated many-body systems, especially their dynamics across quantum phase transitions, are prohibitively difficult to be numerically simulated. We experimentally demonstrate that such complex many-body dynamics can be efficiently studied in a 3D spinor Bose-Hubbard model quantum simulator, consisting of antiferromagnetic spinor Bose-Einstein condensates confined in cubic optical lattices. We find dynamics and scaling effects beyond the scope of existing theories at superfluid-insulator quantum phase transitions, and highlight spin populations as a good observable to probe the quantum critical dynamics. Our data indicate that the scaling exponents are independent of the nature of the quantum phase transitions. We also conduct numerical simulations in lower dimensions using time-dependent Gutzwiller approximations, which qualitatively describe our observations. Three-dimensional (3D) strongly correlated many-body systems and their dynamics across quantum phase transitions pose a challenge when it comes to numerical simulations. The authors experimentally demonstrated that such many-body dynamics can be efficiently studied in a 3D spinor Bose-Hubbard model quantum simulator, and observed dynamics and scaling effects beyond the scope of existing theories at superfluid-insulator quantum phase transitions.