We give new and general sufficient conditions for a Gaussian upper bound on the convolutions Km₊n *Km₊n₋₁ * · · · *Km₊1 of a suitable sequence K₁,K₂,K₃, . . . of complex-valued functions on a unimodular, compactly generated locally compact group. As applications, we obtain Gaussian bounds for convolutions of suitable probability densities, and for convolutions of small perturbations of densities.