This paper is the third and last part of a work attempting to give a unified analysis of discontinuous Galerkin methods. The purpose of this paper is to extend the framework that has been developed in Part II for two-field Friedrichs' systems associated with second-order PDEs. We now consider two-field Friedrichs' systems with partial $L^2$-coercivity and three-field Friedrichs' systems with an even weaker $L^2$-coercivity hypothesis. In particular, this work generalizes the discontinuous Galerkin methods of Part II to compressible and incompressible linear continuum mechanics. We also show how the stabilizing parameters of the method must be set when the two-field Friedrichs' system is composed of terms that may be of different magnitude, thus accounting, for instance, for advection-diffusion equations at high Peclet numbers.