We analytically and numerically investigate the dynamical properties of the tilted dispersion relativistic quasiparticles emerged in a cold atomic optical lattice system. By introducing the next nearest neighboring (NNN) hopping term into Su–Schrieffer–Heeger (SSH) model, the Dirac quasiparticles with tilted dispersion relation are realized. The results show that the tilted dispersion causes a drift in relativistic quasiparticles rather than affecting interference behavior between inner states. To be specific, the relativistic phenomena of the quasiparticles induced by the inner state interference (such as Zitterbewegung, Klein paradox, etc.) is completely unaffected by the tilted dispersion. In order to distinguish the drift induced by tilted dispersion and common initial velocity, we calculate the momentum distribution of the relativistic quasiparticles. We obtain the difference between the drift induced by initial velocity and tilted dispersion. The former affects the ZB, while the latter does not. By using this character, we propose a quench dynamics scheme to obtain a stable mono-spin state. The proposed cold atomic lattice system would provide a promising platform in exploring the intrinsic exotic physics of relativistic quasiparticles and the related systems.