We investigate Floquet engineering of three-dimensional Dirac fermions driven by propagating waves, identifying distinct quantum states and phase transitions in the time-like, light-like, and space-like regimes. Notably, we uncover a novel regime where Floquet Weyl bands emerge and transition into Type-II Weyl states as the wave speed nears the Fermi velocity. Using Floquet-Bloch theory, we demonstrate that Lorentz contraction strongly amplifies Floquet band modulation effects, leading to a shockwave-like state synchronized with the wave motion. These findings extend beyond electrons to quasiparticles with relativistic dispersions, opening new avenues for dynamic band engineering in quantum materials.