Gaining a comprehensive understanding of the characteristics and propagation of precipitation-based meteorological drought to terrestrial water storage (TWS)-derived hydrological drought is of the utmost importance. This study aims to disentangle the frequency–time relationship between precipitation-derived meteorological and TWS-based hydrological drought from June 2002 to June 2017 based on the Standardized Precipitation Index (SPI) and Standardized Terrestrial Water Storage Index (STI) by employing wavelet coherence rather than a traditional correlation coefficient. The possible influencing factors on drought propagation in 28 regions across the world are examined. The results show that the number of drought months detected by the STI is higher than that detected by the SPI worldwide, especially for slight and moderate drought. Generally, TWS-derived hydrological drought is triggered by and occurs later than precipitation-based meteorological drought. The propagation characteristics between meteorological and hydrological droughts vary by region across the globe. Apparent intra-annual and interannual scales are detected by wavelet analysis in most regions, but not in the polar climate region. Drought propagation differs in phase lags in different regions. The phase lag between hydrological and meteorological drought ranges from 0.5 to 4 months on the intra-annual scale and from 1 to 16 months on the interannual scale. Drought propagation is influenced by multiple factors, among which the El Niño–Southern Oscillation, North Atlantic Oscillation, and potential evapotranspiration are the most influential when considering one, two, or three factors, respectively. The findings of this study improve scientific understanding of drought propagation mechanisms over a global scale and provide support for water management in different subregions.