The propagation speed of an auto-ignitive dimethyl-ether (DME)/air mixture at elevated pressures and subjected to monochromatic temperature oscillations is numerically evaluated in a one-dimensional statistically stationary configuration using fully resolved numerical simulations with reduced kinetics and transport. Two sets of conditions with temperatures within and slightly above the negative temperature coefficient (NTC) regime are simulated to investigate the fundamental aspects of auto-ignition and flame propagation along with the transition from auto-ignitive deflagration to spontaneous propagation regimes under thermal stratification. Contrary to the standard laminar flame speed, the steady propagation speed of an auto-ignitive front is observed to scale proportionally to its level of upstream reactivity. It is shown that this interdependence is primarily influenced by the characteristic residence time and the homogeneous auto-ignition delay. Furthermore, the unsteady reaction front in either of the two cases responds distinctly to the imposed stratification. Specifically, the results in both cases show that the dynamic flame response depends on the mean temperature at the flame base Tb and the time-scale of thermal stratification. It is also found that, based on Tb and the propensity of the mixture to two-stage chemistry, the instantaneous peak propagation speed and the overall time taken to achieve that speed differs considerably. A displacement speed analysis is carried out to elucidate the underlying combustion modes that are responsible for such a variation in flame response.