Spatiotemporal optical vortex (STOV) light is a new type of vortex light with transverse orbital angular momentum (OAM) which is different from conventional spatial vortex light. Understanding the properties of STOV are meaningful before STOV are applied. We present a theoretical study on the generation and propagation of spatiotemporal vortices step by step based on diffraction theory. The properties of the output pulses with different topological charges generated using 4 f pulse shaper in both the near-field and the far-field are analyzed. Using spiral phase mask, the intensity profiles of the output pulses immediately after the 4 f pulse shaper are of multi-lobe structures. With energies circulating around the phase singularity in the space-time plane, energy coupling occurs between the spatial and temporal domains in the wave packets during propagation, then the intensity profiles evolve into multi-hole shapes, and the holes tend to be merged for higher order STOV. The conservation of OAM in the space-time domain is shown clearly. The profiles of the output pulses in the near-field form donut rectangle shapes using π-step mask, and in the far-field, they split into a multi-lobe structure. The rules of the generation and evolution of STOV are revealed. The results demonstrate the physical properties of the STOV and the generation and propagation processes directly and clearly. It provides a guidance on the application of STOV.