The impact of geopolitical risk on financial markets has been well-documented, yet there is limited research on higher-order moment risk spillovers in energy markets. This paper employs an extended joint TVP-VAR spillover index model to investigate the risk spillover effects and their dynamic evolution across global energy markets under the moments of volatility, skewness, and kurtosis. By constructing a GARCH-MIDAS-GPR model, we examine the impact of geopolitical risks on the higher-order moment risk connectivity of the energy system. Our findings reveal that: (1) energy markets across countries exhibit strong interconnections in terms of volatility, skewness, and kurtosis, while the spillover effects of skewness and kurtosis are lower than those of conditional variance; (2) higher-order moment risk spillovers exhibit significant time-varying characteristics, tending to substantially increase during periods of geopolitical crises; (3) the major countries in Europe and America act as net exporters of risk at various moment levels, and the Asia-Pacific countries are primarily net importers of risk; (4) the higher-order moment risk spillover indices significantly improve the predictive capabilities for energy market risks; (5) geopolitical risks are a significant factor exacerbating the spillover of higher-order moment risks in energy markets, with an even greater impact on risk spillover at the level of skewness. Our research contributes to a deeper understanding of the risk spillover effects of geopolitical risks on energy markets under higher-order moments, which is of great significance for investors to optimize their portfolios and for regulatory authorities to establish regulatory systems.