The results of a study of ultra-rapid (flash) sintering of oxide ceramic materials under microwave heating with high absorbed power per unit volume of material (10–500 W/cm3) are presented. Ceramic samples of various compositions—Al2O3; Y2O3; MgAl2O4; and Yb(LaO)2O3—were sintered using a 24 GHz gyrotron system to a density above 0.98–0.99 of the theoretical value in 0.5–5 min without isothermal hold. An analysis of the experimental data (microwave power; heating and cooling rates) along with microstructure characterization provided an insight into the mechanism of flash sintering. Flash sintering occurs when the processing conditions—including the temperature of the sample; the properties of thermal insulation; and the intensity of microwave radiation—facilitate the development of thermal runaway due to an Arrhenius-type dependency of the material’s effective conductivity on temperature. The proper control over the thermal runaway effect is provided by fast regulation of the microwave power. The elevated concentration of defects and impurities in the boundary regions of the grains leads to localized preferential absorption of microwave radiation and results in grain boundary softening/pre-melting. The rapid densification of the granular medium with a reduced viscosity of the grain boundary phase occurs via rotation and sliding of the grains which accommodate their shape due to fast diffusion mass transport through the (quasi-)liquid phase. The same mechanism based on a thermal runaway under volumetric heating can be relevant for the effect of flash sintering of various oxide ceramics under a dc/ac voltage applied to the sample.