At the Earth's low-latitude magnetopause, the Kelvin-Helmholtz instability (KHI), driven by the velocity shear between the magnetosheath and magnetosphere, has been frequently observed during northward interplanetary magnetic field (IMF) periods. However, the signatures of the KHI have been much less frequently observed during southward IMF periods, and how the KHI develops under southward IMF has been less explored. Here, we performed a series of realistic 2D and 3D fully kinetic simulations of a KH wave event observed by the Magnetospheric Multiscale (MMS) mission at the dusk-flank magnetopause during southward IMF on September 23, 2017. The simulations demonstrate that the primary KHI bends the magnetopause current layer and excites the Rayleigh-Taylor instability (RTI), leading to penetration of high-density arms into the magnetospheric side. This arm penetration disturbs the structures of the vortex layer and produces intermittent and irregular variations of the surface waves which significantly reduces the observational probability of the periodic KH waves. The simulations further demonstrate that in the non-linear growth phase of the primary KHI, the lower-hybrid drift instability (LHDI) is induced near the edge of the primary vortices and contributes to an efficient plasma mixing across the magnetopause. The signatures of the large-scale surface waves by the KHI/RTI and the small-scale fluctuations by the LHDI are reasonably consistent with the MMS observations. These results indicate that the multi-scale evolution of the magnetopause KH waves and the resulting plasma transport and mixing as seen in the simulations may occur during southward IMF.