A beam of ultrarelativistic charged particles in a plasma can reach equilibrium with its own radial wakefield and then propagate with little change in shape. If some co-moving perturbation appears ahead of the beam, it may or may not destroy the beam with its wakefield, depending on the phase and amplitude of the wakefield. We numerically study which perturbations can destroy a single short bunch or a train of many short bunches at the parameters of interest for plasma wakefield acceleration in an axisymmetric configuration, and how fast. We find that there are particularly dangerous wakefield phases in which the beam can be destroyed by perturbations of very low amplitude. We also find that perturbations with an amplitude larger than the wakefield of a single bunch in the train are always destructive.