Now Mr. Cuellar, 68, has become one of the administration’s most consistent critics on immigration, appearing on Fox News and at times echoing Republicans, saying immigrants are pouring into the United States because they believe “that the border is open.”

His criticism has been met with fierce resistance from Jessica Cisneros, 28, a progressive immigration lawyer who is trying to unseat him in a Democratic runoff on Tuesday.

Like other Democratic primary contests, their race is a proxy battle for the broader direction of a party that is being tugged between moderate and progressive wings. But in particular, it encapsulates the acute tensions within the party on immigration.

In interviews with Democratic leaders and voters in Texas’ 28th Congressional District, which stretches from Laredo to San Antonio, many expressed a deep frustration with both national Democrats and Republicans who use the border as a political backdrop but have failed to overhaul the nation’s immigration laws, combat the drug trade or improve legal pathways to citizenship.