<p>This paper analyzes the conditional mutual information transmission among the main cryptocurrencies and the markets of commodities, stocks, and fiat currencies, providing some insights into the connections structure around various crisis periods. It presents a new conditional mutual information model based on partial correlations, which extends the transfer entropy technique and allows to avoid spurious links. The main findings reveal that the connections are generally weak and that their strength and sign alter during crisis periods. The structure of the connections changed mainly after the 2018 cryptocurrency collapse, when two clusters stood out, one centered on Bitcoin, while the other involved Dogecoin. An additional cluster centered on Litecoin appeared during the COVID-19 pandemic crisis. The Russia-Ukraine war period is characterized by the emergence of a new cluster centered on Ethereum. During the pandemic and war periods, cryptocurrencies acted as safe-havens, offering investors the possibility to hedge their investments against heightened uncertainties associated to these crises. A portfolio analysis underlines the practical implications of the proposed model for improving the portfolio performance.</p>