The relaxational dynamics of weak itinerant magnets below the Curie temperature is investigated. The calculation is based on the non-linear equations of motion for the magnetic order parameter. It is shown that besides the conventional linear relaxation mechanism due to the Landau damping of spin fluctuations (SFS) which gives rise to overdamped paramagnons, there may exist a different mechanism which is due to non-linear three-mode interactions of spin fluctuations. The interplay between these two mechanisms is analysed and it is argued that three-mode scattering with emission, or absorption, of a longitudinal SF by a magnon may dominate. This results in a new type of overdamped longitudinal SF propagating close to the magnon dispersion, which has probably been observed in recent neutron scattering experiments.