[For Parts I and II, see Sib. Math. J. 36, No. 2, 305-323 (1995; Zbl 0876.30023) and Sib. Math. J. 38, No. 2, 281-295 (1997; Zbl 0947.30012).] The author studies stability of the class \(\mathfrak G^2\) of mappings defined as follows. Let \(n\) and \(m\) be naturals such that \(n>m\) \((m\geq 2)\). Then \(\mathfrak G^2\) is the class of all mappings \(g\:\Delta\to\mathbb R^m\) on domains \(\Delta\) in \(\mathbb R^n=\mathbb R^m\times\mathbb R^{n-m}\) for each of which there exists either an orientation preserving Möbius transformation or a constant mapping \(g_0\:\mathbb R^m\to\mathbb R^m\) such that the equality \(g(t_1,\dots ,t_n)=g_0(t_1,\dots ,t_m)\) holds for all \((t_1,\dots ,t_n)\in\Delta\). The aim of this article is threefold: (1) to complete the proof of the assertion that classes \(\mathfrak G^2\) are stable over compact subsets of a ball but not over the whole ball; (2) to study properties of mappings which are close to the class \(\mathfrak G^2\); (3) to discuss some observations that, from the author's viewpoint, are important for the theory of \(\xi\)-stability and its generalizations.