The atmospheric temperature distribution is typically described by its mean and variance, while higher order moments, such as skewness, have received less attention. Skewness is a measure of the asymmetry between the positive and negative tails of the distribution, which has implications for extremes. It was recently shown that near-surface temperature in the southern hemisphere is positively skewed on the poleward side of the storm tracks, and negatively skewed on the equatorward side. Here we take a dynamical approach to further study what controls the spatial structure of the near-surface temperature distribution in this region. We employ a tracking algorithm to study the formation, intensity, and movement of warm and cold temperature anomalies. We show that warm anomalies are generated on the equatorward side of the storm tracks and propagate poleward, while cold anomalies are generated on the poleward side and propagate equatorward. We further show that while the perturbation growth is mainly achieved through linear meridional advection, it is the nonlinear meridional advection that is responsible for the meridional movement of the temperature anomalies and therefore to the differential skewness. The projected poleward shift and increase of the temperature variance maximum in the southern hemisphere under global warming is shown to be composed of a poleward shift and increase in the maximum intensity of both warm and cold anomalies, and a decrease in their meridional displacements. An analytic expression is derived for the nonlinear meridional temperature tendency, which captures the spatial structure of the skewness and its projected changes.