As a follow up to papers dealing firstly with a convective variational formulation in a Milne-Cartan framework for non-dissipative multi-fluid models, and secondly with various ensuing stress energy conservation laws and generalized virial theorems, this work continues a series showing how analytical procedures developed in the context of General Relativity can be usefully adapted for implementation in a purely Newtonian framework where they provide physical insights that are not so easy to obtain by the traditional approach based on a 3+1 space time decomposition. The present paper describes the 4-dimensionally covariant treatment of various dissipative mechanisms, including viscosity in non-superfluid constituents, superfluid vortex drag, ordinary resistivity (mutual friction) between relatively moving non-superfluid constituents, and the transvective dissipation that occurs when matter is transformed from one constituent to another due to chemical disequilibrium such as may be produced by meridional circulation in neutron stars. The corresponding non-dissipative limit cases of vortex pinning, convection and chemical equilibrium are also considered.