Compared to earthquakes and volcanoes, drought is one of the most damaging natural disasters and is mainly affected by rainfall losses, especially by the runoff regulation ability of the underlying watershed surface. Based on monthly rainfall runoff data recorded from 1980 to 2020, in this study, the distributed lag regression model is used to simulate the rainfall-runoff process in the karst distribution region of South China, and a time series of watershed lagged-flow volumes is calculated. The watershed lagged effect is analyzed by four distribution models, and the joint probability between the lagged intensity and frequency is simulated by the copula function family. The results show that (1) the watershed lagged effects simulated by the normal, log-normal, P-III and log-logistic distribution models in the karst drainage basin are particularly significant, with small mean square errors (MSEs) and significant time-scale characteristics. (2) Affected by spatiotemporal distribution differences in rainfall and the impacts of different basin media and structures, the lag response of runoff to rainfall differs significantly among different time scales. Especially at the 1-, 3- and 12-month scales, the coefficient of variation (Cv) of the watershed lagged intensity is greater than 1, while it is less than 1 at the 6- and 9-month scales. (3) The lagged frequencies simulated by the log-normal, P-III and log-logistic distribution models are relatively high (with medium, medium–high and high frequencies, respectively), while that simulated by the normal distribution is relatively low (medium–low and low frequencies). (4) There is a significant negative correlation (R< − 0.8,Sig. < 0.01) between the watershed lagged intensity and frequency. For the joint probability simulation, the fitting effect of the gumbel Copula is the best, followed by the Clayton and Frank-1 copulas, and while that of the Frank-2 copula is relatively weak. Consequently, the propagation mechanism from meteorological drought to agricultural or hydrological drought and the conversion mechanism between agricultural and hydrological drought are effectively revealed in this study, thereby providing a scientific basis for the rational utilization of water resources and drought resistance and disaster relief in karst areas.