WASHINGTON — With Christine Lagarde, the current leader of the International Monetary Fund, poised to become the next president of the European Central Bank, the big questions on Wednesday were who would succeed her and whether the United States would break with tradition and try to install an American in the post.

On Tuesday, European officials nominated Ms. Lagarde to succeed Mario Draghi as the European Central Bank’s president. Ms. Lagarde, a stabilizing force who has lent star power to the monetary fund since taking over as its managing director in 2011, plans to step away from her responsibilities while the nomination process moves forward.

The expected transition of power at the fund comes as the global economy slows and protectionism increases amid President Trump’s multifront trade war. The next director of the fund will have to confront economic woes in Argentina, Venezuela and Turkey, as well as take over the kind of multilateral institution that Mr. Trump has long criticized as overstepping its authority.

“The key challenge for the next I.M.F. head is how to maintain the institution’s relevance, influence and legitimacy in a world where multilateralism is breaking down amid trade tensions and shifting geopolitical alliances,” said Eswar Prasad, the former head of the monetary fund’s China division. “The I.M.F. is the epitome of multilateralism in international finance, but it suffers from marginalization by the key advanced economies, a pool of resources dwarfed by global capital flows, and lack of trust among emerging market economies.”