As crypto markets become more integrated, measuring their spillovers with financial markets becomes fundamental for portfolio choice and risk management. We investigate high-order moment transmission between emerging/developed and digital asset markets through a flexible semi-nonparametric approach that accounts for dynamic conditional correlation and spillover effects, not only in conditional volatility but also in conditional skewness and kurtosis. The results show a (positive) transmission of volatility from emerging and developed markets to digital asset markets, as a signal of market integration, but also some positive/negative skewness and kurtosis spillovers (remarkably from Crypto and Blockchain indices to Emerging Asia and Latin America indices) detected at daily and weekly basis.