Recent ionic conductivity and tracer diffusion measurements over a large range of the mobile ion content x, carried out for Ag+- and Cu+-conducting chalcogenide and chalcohalide glasses, show two distinctly different ion transport regimes above the percolation threshold at approximate to 30 ppm M+: (i) a critical percolation regime at low x, and (ii) modifier-controlled ion transport at high x. Using a number of structural and spectroscopic techniques (high-resolution neutron diffraction, small-angle neutron scattering, high-energy X-ray diffraction, EXAFS, I-129-Mossbauer spectroscopy), we will show that the two regimes have a clear structural basis. Transport properties in the critical percolation domain depend almost exclusively on the connectivity of the host matrix represented by the average coordination number : the nature of the mobile cations and chemical form of the dopant or of the host network do not play any important role. In contrast, the connectivity of the cation-related structural units MYz (Y = chalcogen or halide, z = 3 or 4), evidenced by the short M-M correlations (from 2.7 to 4.2 A) and reflected by the M-M coordination number, appears to be predominant in the modifier-controlled region. Highly connected edge- or corner-sharing (ES or CS) MY units, which form at least 2D sheets or tunnels in the glass network, lead to the highest mobility of the M+ ions.