Quenched disorder slows down the scrambling of quantum information. We formulate a kinetic theory of scrambling in a d-dimensional strongly-correlated metal in the vicinity of a superconducting phase, following the scrambling dynamics as the impurity scattering rate is increased. Within this framework, we rigorously show that the butterfly velocity v is bounded by the light cone velocity v lc = vF/ √ d where vF is the Fermi velocity. We analytically identify a disorder-driven dynamical transition occurring at small but finite disorder strength between a spreading of information characterized at late times by a discontinuous shock wave propagating at the maximum velocity v lc , and a smooth traveling wave belonging to the Fisher or Kolmogorov-Petrovsky-Piskunov (FKPP) class and propagating at a slower, if not considerably slower, velocity. In the diffusive regime, we establish the relation v 2 /λFKPP ∼ D el where λFKPP is the Lyapunov exponent set by the inelastic scattering rate and D el is the elastic diffusion constant.