One-dimensional systems often possess multiple channels or bands arising from the excitation of transverse degrees of freedom. In this work, we study the specific processes that dominate the equilibration of multichannel Fermi gases at low temperatures. Focusing on the case of two channels, we perform an analysis of the relaxation properties of these systems by studying the spectrum and eigenmodes of the linearized collision integral. As an application of this analysis, a detailed calculation of the bulk viscosity is presented. The dominant scattering processes obey an unexpected conservation law which is likely to affect the hydrodynamic behavior of these systems.