Quantifying the connectedness across the oil and stock markets is vital for global asset allocation, systemic risk regulation, and prudent policy making. However, existing studies are mostly confined to return (first moment) and volatility (second moment) levels. This paper integrates the GARCHSK model with the TVP-VAR extended joint connectedness approach to examine higher-order moment risk connectedness between international oil and stock markets in both G7 and BRICS countries, thereby encompassing a wide range of global markets. This paper also discusses the impacts of geopolitical risks, acts, and threats on the total spillovers of volatility and higher-order moments based on the wavelet coherence method. Empirical results demonstrate that the risk spillovers between oil and stock markets are moment-dependent with the total kurtosis spillovers being stronger than the volatility and skewness spillovers. WTI, Brent, INE oil futures, and the Chinese and Japanese stock markets are consistent net receivers of spillovers whereas the remaining G7 stock markets are net transmitters. G7 and the Russian stock markets transmit significant net spillovers of higher-order moments to oil futures. The dynamic spillovers of volatility, skewness, and kurtosis are all time-varying and highly sensitive to the outbreak of major crisis events, especially the China-US trade war, the COVID-19 outbreak, and the Russia-Ukraine conflict. Driving factor analysis shows that the impacts of geopolitical risks, acts, and threats on total spillovers vary over time and across various frequency bands. Geopolitical risks strongly impact the total skewness spillovers during the crisis periods. Geopolitical acts exert a stronger influence on total kurtosis spillovers than geopolitical risks or threats. These findings can offer practical implications for cross-market investors, portfolio managers, market regulators, and policymakers.