Basin response depends on the soil properties (site geometry, impedance contrast), on the constitutive model and on the input motion. Numerical modelling is a useful tool to understand the role and the influence of these different parameters governing site effects. In this study, we focus on the 2-D P–SV seismic wave propagation in a simple asymmetric basin model. We assess the influence of the material constitutive model (elastic, viscoelastic, elastoplastic and viscoelastoplastic) on the wave propagation as well as the importance of the input motion on the development of material non-linearity. We also show that both viscoelasticity and material non-linearity strongly modify the ground motion on a broad range of frequencies, yet they have a stronger effect in some particular frequency bands. We advise that realistic simulations should couple both viscoelasticity accounting for energy dissipation at small strains and non-linear soil behaviour taking into account hysteresis energy dissipation and shear modulus degradation at higher strain levels. In addition, we study the influence of the source frequency content, intensity and complexity on the Fourier and response spectra of horizontal acceleration time histories, and the maximum shear stresses and strains computed within the non-linear material filling the basin. We show that the complexity and amplitude of the source determines whether damping or reduction of the S-wave velocity dominates the wave propagation in the non-linear media. Indeed, for a simple impulsive input the high frequencies are damped whereas for a complex input motion, such a real earthquake, high frequency damping comes together with frequency shift to lower frequency values. For this reason, the use of the source peak ground acceleration (PGA) only is not a good indicator to characterize soil non-linear effects because of the simultaneous influence of input motion intensity and complexity, impedance contrast, and material strength. However, testing other ground motion indicators is beyond the scope of this paper. At last, we test the importance of dynamic soil properties on the wave propagation and show that shear modulus and damping ratio curves that are relatively close, regarding uncertainties associated to such curves measurements, may lead to notable variations in the acceleration fields. Simulations should take into account this variability to assess the uncertainties on computed ground motion.