This is the third of a series of papers that presents a kinetic theory of fluctuations in equilibrium classical fluids that makes extensive use of diagrammatic techniques in its development and that will facilitate the use of diagrammatic techniques in the derivation of approximate kinetic theories. The fundamental fluctuating quantity in the theory is f(R,P), the density of particles (atoms) at points in single particle phase space, and the time correlation functions for fluctuations of this quantity from its average are the quantities that the theory is designed to calculate. In this paper, we start with graphical diagrammatic expressions for the correlation functions and for closely related response functions, developed in paper II, and analyze the cluster properties of the various renormalized interactions that appear in the theory. This allows us to derive a second diagrammatic formulation that has many similarities to the Mayer cluster theory for equilibrium correlation function. This second formulation allows us to express the correlation functions, response functions, and various memory functions in a common graphical language that facilitates the derivation of nonlinear relationships between, for example, the memory function and the correlation function. Such relationships are time dependent analogues of the various closures (PY, HNC, RHNC, etc.) used to obtain theories of the equilibrium structure of fluids and are central to various versions of the mode coupling theory of relaxation in fluids.