The increasing use of natural gas as an efficient, reliable, affordable, and cleaner energy source, compared with other fossil fuels, has brought the catalytic CH4 complete oxidation reaction into the spotlight as a simple and economic way to control the amount of unconverted methane escaping into the atmosphere. CH4 emissions are a major contributor to the ‘greenhouse effect’, and therefore, they need to be effectively reduced. Catalytic CH4 oxidation is a promising method that can be used for this purpose. Detailed studies of the activity, oxidative thermal aging, and the time-on-stream (TOS) stability of pristine La1−xSrxMnO3 perovskites (LSXM; X = % substitution of La with Sr = 0, 30, 50 and 70%) and iridium-loaded Ir/La1−xSrxMnO3 (Ir/LSXM) perovskite catalysts were conducted in a temperature range of 400–970 °C to achieve complete methane oxidation under excess oxygen (lean) conditions. The effect of X on the properties of the perovskites, and thus, their catalytic performance during heating/cooling cycles, was studied using samples that were subjected to various pretreatment conditions in order to gain an in-depth understanding of the structure–activity/stability correlations. Large (up to ca. 300 °C in terms of T50) inverted volcano-type differences in catalytic activity were found as a function of X, with the most active catalysts being those where X = 0%, and the least active were those where X = 50%. Inverse hysteresis phenomena (steady-state rate multiplicities) were revealed in heating/cooling cycles under reaction conditions, the occurrence of which was found to depend strongly on the employed catalyst pre-treatment (pre-reduction or pre-oxidation), while their shape and the loop amplitude were found to depend on X and the presence of Ir. All findings were consistently interpreted, which involved a two-term mechanistic model that utilized the synergy of Eley–Rideal and Mars–van Krevelen kinetics.