Abstract: The movement and merging of Dirac points were realized in the Su-Schrieffer-Heeger (SSH) model by adjusting short-range interaction. This process corresponded to a topological phase transition from a semi-metallic to a band insulating phase, known as Lifshitz phase transition. The dynamical properties of relativistic quasi-particles in this process were investigated with the analytical and numerical methods. The system exhibited relativistic dynamics under the condition of weak short-range interaction (i.e., before the merging of Dirac points). However, with the increase of the short-range interaction, the merging of Dirac points occurred. After that the system showed non-relativistic dynamics. Therefore, the phase transition was also a transition from relativistic to nonrelativistic dynamics. Furthermore, through numerical simulation, a vivid evolution of quasi-particles' density distribution versus time was presented before (relativistic) and after (non-relativistic) the merging transition. The results reveal that before the phase transition the monochromatic Dirac quasi-particles split while the bi-chromatic Dirac quasi-particles directionally drift. Furthermore, after the phase transition, no matter how the initial state changes, no component splitting occurs in the system. Finally, the centroid motion curves (worldlines) of quasi-particles under different interaction conditions are provided.