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Wave propagation in magnetically structured atmospheres is a thoroughly studied, yet practically inexhaustible well of investigations in the field of solar magneto-seismology. A simple but powerful example is the examination of wave behaviour in a magnetic slab. Our previous study (Zsámberger, Allcock and Erdélyi, Astrophys. J., 853, p. 136, 2018) used an analytical approach to derive the general dispersion relation for magneto-acoustic waves in a magnetic slab of homogeneous plasma, which was enclosed in an asymmetric magnetic environment. In the present study, we focus on the analysis of wave propagation in various limiting cases applicable to solar and space plasma or astrophysics. The thin- and wide-slab approximations, as well as the limits of low and high plasma-beta values are considered. Utilising the fact that in a weakly asymmetric slab, the dispersion relation can be decoupled, the behaviour of quasi-sausage and quasi-kink modes is studied in further analytical and numerical detail, and their avoided crossings are described. The results highlight how the asymmetry influences the wave properties, e.g. the phase speed of eigenmodes, depending on the ratios of external-to-internal densities and magnetic fields on the two sides. Notably, the phase speeds of surface modes will converge to different values for quasi-sausage and quasi-kink modes in the wide-slab limit, and cut-off frequencies are introduced with respect to both surface and body modes, in thin as well as wide slabs, beyond which the solutions become leaky. These obtained properties of MHD wave behaviour could be measured with suitable high-resolution instruments in the future.