Nearly a year since Texas adopted a law empowering state and local police officers to arrest undocumented migrants who cross into its territory, Republican lawmakers in at least 11 states have tried to adopt similar measures, capitalizing on the prominence of immigration in the 2024 presidential election.

The fate of the proposals — six have been enacted or are under consideration, with Louisiana expected to sign its measure into law as early as next week — is still being litigated. In a case before a federal appeals court, Texas is defending its law by arguing that illegal immigration is a form of invasion, allowing it to expand its power to protect its borders. Federal courts have previously ruled that, from a constitutional perspective, the definition of the term invasion is limited to military attacks.

States have tested the limits of their power over immigration before, but lawyers and legal scholars said the push this year was accompanied by what had amounted to a public-relations campaign.

In campaign speeches, political ads and the halls of Congress, more Republicans are echoing former President Donald J. Trump by arguing that the rise of migration at the southern border is an “invasion.” President Biden, under pressure from both Republicans and Democrats to tackle the issues at the border, signed an executive order this month to curb asylum, and he could have more actions coming next week.

The measure expected to be signed by Gov. Jeff Landry, Republican of Louisiana includes provisions allowing Mr. Landry and his attorney general to establish a compact with Texas to address border security. Mr. Landry has already met with Gov. Greg Abbott, Republican of Texas, and dispatched Army National Guard soldiers from Louisiana to Texas’ border with Mexico.

Valarie Hodges, the state senator in Louisiana who wrote the legislation, joined other Republicans in calling Mr. Biden’s recent action “too little, too late,” saying in an interview that state measures like hers were essential because the Biden administration had failed to enforce immigration laws.

“The federal government is not helping us,” she said. “They have done the opposite — they have flung open the doors and let more people in.”

In the swing state of Arizona, Republican lawmakers this month put a Texas-style measure on the ballot in November, after their state’s Democratic governor, Katie Hobbs, vetoed similar legislation. And in Michigan, another battleground where immigration has deeply energized Mr. Trump’s base, Republican state lawmakers with the far-right Freedom Caucus introduced yet another measure.

James DeSana, a state representative in Michigan, said he and the bill’s other authors decided to file it after a visit to Del Rio and Eagle Pass, Texas, though they believe it will most likely stall in the Democratic-controlled State Legislature.

Mr. DeSana, a Republican who campaigned against “sanctuary” cities when he won his seat — and flipped it from Democratic control — in 2022, emphasized that he was not against legal immigration or creating more temporary legal pathways for workers into the country. But he was firm in his view that the situation at the southern border had become an invasion.

“A lot of people end up in inner cities,” he said in an interview. “We don’t have enough housing. Our police resources are stressed. Crimes are being committed.”

Democrats, immigrant-rights groups and some legal scholars said the proposals could devastate their states’ economies, lead to racial and ethnic profiling and advance dangerous visions of undocumented immigrants as hostile invaders and aliens. Arizona’s ballot measure has stirred memories of police harassment and anti-immigrant sentiment among young Latino and immigrant rights activists who have successfully pushed back against such restrictive immigration laws before.

On the floor of the Louisiana House in April, State Senator Royce Duplessis, a Democrat from New Orleans, urged lawmakers in his state and nationwide to reject language that spurred images of undocumented immigrants as “though they’re coming from outer space to take us all out of our homes.”

In an interview, he said states with fewer resources were unlikely to fare better than the federal government in handling immigration, a complex issue that both parties had failed to address for years. “It is pushing an ideological agenda more than addressing real issues of public safety,” he said.

Texas has been experimenting with pushing the limits of its powers on hot-button issues other than immigration, including abortion and gender-transition restrictions, but its campaign has gained the most traction with immigration.

Mr. Abbott’s busing of migrants to blue cities like New York and Chicago at first drew condemnation from immigrant-rights groups and progressives, who argued that he was treating migrants like political pawns — and then concern, including among Democrats, that local and state governments were unequipped to handle the record levels of migration under the Biden administration.

Proponents of the state measures contend that a 1996 federal law to curb illegal immigration enhanced states’ abilities to help with immigration enforcement, even as the power to regulate immigration and naturalization lies with Congress. But efforts to broaden the powers of law enforcement to carry out immigration laws in the decades since have largely been curbed by the courts. Federal judges blocked key aspects of immigration laws adopted in Arizona in 2010 and in South Carolina in 2011, including provisions that required law enforcement officers to check some people’s immigration status on routine stops, and immigrants to carry federal registration documents.

In committee hearings and floor debates more recently, Republicans have stressed that their descriptions of an invasion at the southern border are accurate, pointing to the flow of fentanyl across the border and to cases of human trafficking, murders and sexual assaults committed by undocumented immigrants.

According to the Drug Enforcement Administration, a vast majority of fentanyl in the United States is smuggled through legal ports of entry, typically by citizens driving across the border, and though the immigrant population in the country has been growing for decades, crime in the same period has decreased.

In the Texas case before the federal appeals court, Ilya Somin, a professor at the Antonin Scalia Law School at George Mason University in Virginia, argued in an amicus brief on behalf of himself and the Cato Institute, a libertarian research center, that expanding the definition of invasion to include illegal immigration would set a dangerous precedent, enabling states to declare war with foreign powers whenever they desired and to detain more people without due process, regardless of citizenship.

“It goes against the text and original meaning of the Constitution” and will have dire implications, Mr. Somin said in an interview.

Jennifer M. Chacón, a professor at Stanford Law School who researches immigration and constitutional law, said rhetoric in the Texas case stirring fears of immigrant invasions had cropped up throughout the nation’s history, playing into harmful racial and ethnic tropes and bigotry.

“An invasion envisions an armed group that is acting cohesively to enact an act of war and merits a response. That is not what it is,” she said, referring to an increase in immigration all over the world. “This is a multinational group of men, women and children who are fleeing for a variety of reasons.”

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