Droughts, extremely destructive natural disasters, pose a significant threat to water resource management, ecosystem balance, and socioeconomic development because of their variable spatial dynamics and complex propagation processes. In this study, temperature, precipitation, runoff, and soil moisture data from the ERA5-Land product were employed to estimate the Standardized Precipitation Evapotranspiration Index (SPEI), Standardized Runoff Index (SRI), and Standardized Soil Moisture Index (SSMI) at different time scales, which can represent meteorological, hydrological, and agricultural droughts, respectively. The cross-wavelet transform, the maximum Pearson correlation coefficient method, and the geodetector model were used to analyze the evolutionary characteristics, propagation processes, and drivers of these drought types on the Mongolian Plateau from 1982 to 2021. The results revealed the following: (1) Meteorological, hydrological, and agricultural droughts shifted from wetting to drying trends at different time scales. (2) There were significant positive correlations between the SPEI&SRI, SPEI&SSMI, and SRI&SSMI across various time-frequency periods, with most phase arrows pointing down to the right, thus suggesting a meteorological-hydrological-agricultural drought propagation process. (3) The response times of the SRI to the SPEI at the different time scales were 7, 5, 3, and 3 months, whereas for the SSMI, the response times were 7, 6, 6, and 5 months, respectively. The response time of agricultural drought to meteorological drought was longer than that of hydrological drought. (4) Spatially, the average response times of the SRI and SSMI to SPEI were 4.72 and 5.30 months, respectively, whereas the response time of the SRI to SSMI was 2.11 months. In hyper-arid, arid, and semi-arid regions, the drought propagation process was meteorological-hydrological-agricultural; in sub-humid and humid regions, it was meteorological-agricultural-hydrological. (5) Precipitation and temperature significantly influenced the spatiotemporal heterogeneity in the propagation from meteorological drought to hydrological and agricultural droughts, while wind speed was a crucial factor driving the propagation from hydrological to agricultural droughts. These findings provide a theoretical basis for understanding the drought propagation mechanism, establishing a drought warning system, and improving comprehensive drought resistance.