GUADALUPE, Mexico — As an election to choose a union at a Mexican tire plant began one recent drizzly morning, a labor leader urged supporters to come out and vote: “You can relax,” he said over social media. “Your vote is free and secret.”

If he had emphasized the secrecy of the vote it was because, for decades, Mexican workers had little say in choosing the unions that signed contracts with employers in their name. Instead, governments granted their allies in the union movement control over labor.

Now the government of the left-wing president, Andrés Manuel López Obrador, has promised to upend this system.

A new law that guarantees workers the right to decide who will represent them took effect last month. But independent labor leaders say that transforming a decades-old system — overcoming opposition from employers and powerful, politically favored unions, along with deep suspicion by workers — is slow and arduous work.