We study the dynamics of perturbations around non-thermal fixed points related to universal scaling phenomena in quantum many-body systems far from equilibrium. For N-component scalar quantum field theories in 3+1 spacetime dimensions, we use self-consistent large N-extensions to determine the next-to-next stability scaling exponents. Our analysis reveals the presence of both stable and unstable perturbations, the latter leading to subexponential deviations from a fixed point in the infrared. By computing spectral functions, we identify the towers of the far-from-equilibrium quasiparticle states and their dispersion relations. With the help of linear response theory, we show that unstable dynamics arise from competition between elastic scattering processes between quasiparticle states. What ultimately makes fixed points dynamically attractive with nonzero momentum is the universal scaling of the unstable region to the infrared by self-similar quasiparticle cascades. Our results provide an ab initio understanding of the imperative stability properties in self-organized scaling phenomena.