We demonstrate the existence and curious propagation dynamics of self-accelerating beams in Λ-type three-level nonlinear atomic vapors with Kerr and cubic-quintic nonlinearities through numerical investigation and mathematical modelling. Upon adjusting the generation and propagation conditions, these nonlinear accelerating beams exhibit different evolution properties. We demonstrate that the input beams can propagate robustly in the medium regardless of its absorption properties. The shape and peak intensity of the main lobes of these input beams, which are the eigenmodes of the nonlinear Schrödinger equation (NLSE) in atomic media, are preserved for a significantly long propagation distance. If such beams are not the modes of the system, they are subjected to the under-healing or over-healing effect, which damages the shape of the self-accelerating beams. We also discuss the interactions between truncated accelerating beams, which readily generate non-accelerating solitons and soliton pairs. The results indicate that the atomic vapor can serve as a promising medium for the generation of nonlinear self-accelerating beams.