We theoretically study the propagations and interactions of an Airy beam, soliton, and nonlinear self-accelerating beam in LiNbO3 crystal modulated by a shorter wavelength background beam. Simulations indicate that the self-accelerating beam and Airy beam can effectively propagate even in self-defocusing media, and the finite-energy Airy beam can generate bright solitons in a self-focusing medium. When two counter-shift Airy beams interact under appropriate conditions, fusion solitons, breathing solitons, and soliton pairs are generated. If we allow a photovoltaic soliton to collide with an Airy beam, X-type waves and soliton deflection occur at a specific phase shift and beam interval. Moreover, we find that the dynamics of two truncated nonlinear self-accelerating beams tend to exhibit straight propagation with attraction and repulsion in the in- and out-of-phase cases, respectively. These results are significant for applications related to optical routing and optical switches.