We discuss general features of charge transport in nonrelativistic classical field theories invariant under non-Abelian unitary Lie groups by examining the full structure of two-point dynamical correlation functions in grand-canonical ensembles at finite charge densities (polarized ensembles). Upon explicit breaking of non-Abelian symmetry, two distinct transport laws characterized by dynamical exponent z=2 arise. While in the unbroken symmetry sector, the Cartan fields exhibit normal diffusion, the transversal sectors governed by the nonlinear analogs of Goldstone modes disclose an unconventional law of diffusion, characterized by a complex diffusion constant and undulating patterns in the spatiotemporal correlation profiles. In the limit of strong polarization, one retrieves the imaginary-time diffusion for uncoupled linear Goldstone modes, whereas for weak polarizations the imaginary component of the diffusion constant becomes small. In models of higher rank symmetry, we prove absence of dynamical correlations among distinct transversal sectors.