A recently proposed nonlinear transport equation is used to model bulk viscous cosmologies that may be far from equilibrium, as happens during viscous fluid inflation or during reheating. The asymptotic stability of the de Sitter and Friedmann solutions is investigated. The former is stable for bulk viscosity index q < 1 and the latter for q > 1. New solutions are obtained in the weakly nonlinear regime for q = 1. These solutions are singular and some of them represent a late-time inflationary era.