VILNIUS, Lithuania—A week after Russia’s leading opposition figure Alexei Navalny died in an Arctic prison colony in February, his wife, Yulia Navalnaya, met with his grieving aides to ask: What next? It was a pressing question not just for an organization that, despite Navalny’s star power, had struggled to dent President Vladimir Putin’s regime, but for the Russian opposition as a whole. Russia’s invasion of Ukraine in 2022 and Putin’s brutal crackdown on dissent in its wake have scattered the political opposition to the winds, with disparate factions setting up in Western capitals. Former energy tycoon Mikhail Khodorkovsky helps coordinate...