We develop a covariant formalism to study nonlinear perturbations of dissipative and interacting relativistic fluids. We derive nonlinear evolution equations for various covectors defined as linear combinations of the spatial gradients of the local number of e-folds and of some scalar quantities characterizing the fluid, such as the energy density or the particle number density. For interacting fluids we decompose perturbations into adiabatic and entropy components and derive their coupled evolution equations, recovering and extending the results obtained in the context of the linear theory. For non-dissipative and noninteracting fluids, these evolution equations reduce to the conservation equations that we have obtained in recent works. We also illustrate geometrically the meaning of the covectors that we have introduced.