For pt.II see ibid., vol.2, p.7387 (1990). The Fm3m to or from R3 phase transition of MIIZrF6 fluorides (MII=Co,Zr) is analysed by means of a microscopic model, developed in the framework of Landau theory, including two coupled order parameters, namely the elastic strain with F2g symmetry, and the MII(Zr)F6 octahedra rotation coordinate. The model is able to account for the first-order type of this transition, together with its 'improper' ferroelastic character. In addition, expressions are derived to account for the static and dynamic properties of the order parameters, which are able to reproduce consistently the experimental data obtained in the previous two papers.