OTTAWA — The race to replace Canadian Prime Minister Justin Trudeau is dominated by one name: Donald Trump. How to wrestle with the incoming president and his tariff threats has emerged as the defining question in the Liberal Party leadership contest.

OTTAWA—Mark Carney, the former Canadian and U.K. central banker, formally declared his interest in succeeding Justin Trudeau as leader of Canada’s Liberal Party and the country’s prime minister, promising an economic agenda focused on lifting it from a period of stagnant growth.