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Two preceding papers (Parts I and II) present a generalised system of equations to describe non-turbulent flows of inclusion-fluid mixtures. The two first order equations for each phase correspond to equations of the standard two-fluid models. The closure problem, consisting in the derivation of constitutive laws for unknown quantities (mainly interaction terms between phases) is not encountered in our approach; these quantities are provided by two infinite sequences of higher order equations controlling more and more conditioned "disturbance fields". The difficulty is then to truncate in a consistent way these sequences using a diluteness assumption. Agitation terms are also unknowns, both in first and higher orders equations. These terms, which result from various types of micro-motions of both phases (the so-called pseudo-turbulence phenomena), have not been related to disturbance fields at the end of the two preceding papers; they just came out as unclosed correlation functions. This paper gives each correlation function a specific expression; some of them must be approximated due to diluteness in order to be effectively computed.