Disc instabilities such as arm and bar formation, minor mergers and tidal encounters drive a galaxy from equilibrium. Using the theory that describes the evolution of a galaxy halo as a result of stochastic fluctuations developed in the companion paper to this one, we show that this sort of noise evolves a halo toward a standard profile, independent of its initial profile and concentration. This process can substantially redistribute the mass in dark-matter haloes in the 10 Gyr since formation. Three different noise processes are studied: (i) a bombardment by blobs of mass that are small compared to the halo mass (‘shrapnel’); (ii) orbital evolution of substructure by dynamical friction (‘satellites’) and (iii) noise caused by the orbit of blobs in the halo (‘black holes’). The power spectrum in the shrapnel and satellite cases is continuous and produces the universal form by exciting the same discrete modes independent of the noise source. These modes dominate the evolution of the mass profile. The power spectrum for black holes is discrete and has a different form with a much slower rate of evolution. The predicted convergence in evolution may help explain the similarity of galaxy properties in widely differing environments.