TIJUANA, Mexico — Pushed beyond their limits by prolonged waits in dangerous and squalid conditions in parts of Northern Mexico, thousands of caravan members who had been waiting to seek asylum in the United States appear to have given up, Mexican officials said, dealing President Trump an apparent win after a humbling week for his immigration agenda.

About 6,000 asylum seekers who had traveled en masse, many of them in defiance of Mr. Trump’s demands that they turn around, arrived in Northern Mexico in late November as part of a caravan that originated in Honduras. Since then, more than 1,000 have accepted an offer to be returned home by the Mexican government, the officials said. Another 1,000 have decided to stay in Mexico, accepting work permits that were offered to them last fall, at the height of international consternation over how to deal with the growing presence of migrant caravans.

Mr. Trump resorted on Friday to declaring a national emergency after he failed to secure funding from Congress for a border wall that he said would block migrants from entering the United States. But the data from Mexican officials suggested that harsh policies he has introduced to crack down on asylum seekers may already be achieving some of its intended effects.

Added this week to new policies that are bearing down on asylum seekers — which include tight limits on the number of people who can apply for the status each day and a heightened standard of proof to qualify — was the extension of a rule that certain asylum seekers must wait in Mexico for the full duration of their legal cases, which can take years.