The complexity of carbon market mechanisms and the uncertainty in market conditions raise questions on how carbon and energy markets interact. The majority of existing studies focused on the lower-order moment spillover across carbon and energy markets, thereby posing limitations on carbon risk management and carbon market efficiency. This paper analyzes multiscale skewness and kurtosis spillovers across carbon and energy markets under different market conditions. It is found that carbon market is a short-term net skewness spillover receiver, but becomes a medium-term risk source under all market conditions, capable of transmitting skewness risk to the natural gas market. The carbon market acts as a short- and medium-term kurtosis risk source for the natural gas and electricity markets in the lower probability of extreme returns, but bears the kurtosis risk from the natural gas and coal markets when the extreme risk is high. These results indicate that policymakers should take measures to adjust carbon prices when facing long-term skewness risk and a lower probability of extreme returns in the carbon market, to prevent further spread of risk to the energy markets.