Poly[bis(methoxy)phosphazene] (PBMP) is used as a model to investigate the backbone dynamics of polyphosphazenes approaching their glass transitions. Specifically, we study PBMP featuring deuterated methyl groups so that, as a consequence of fast rotation of the methyl groups, H-2 NMR probes the reorientation of their 3-fold symmetry axes and, thus, of the inorganic backbone. Combining H-2 NMR spin-lattice, relaxation, line-shape, and stimulated-echo analyses, we follow the slowdown of the segmental motion upon cooling over a broad temperature/time range. Comparison of present and previous results provides no evidence that polymers featuring inorganic and organic backbones, respectively, show fundamentally different dynamical behaviors during vitrification. In particular, typical of glass-forming polymer melts, we find for the alpha-process of PBMP that its temperature dependence deviates from an Arrhenius law and its time dependence differs from a single-exponential function. H-2 NMR three-time correlation functions indicate that both homogeneous and heterogeneous dynamics contribute to the nonexponential relaxation. In addition. H-2 NMR spin-lattice relaxation and line-shape analyses reveal the existence of some large-angle anisotropic precursor motion in the moderately viscous melt, which may be a peculiarity of polyphosphazene.