For many months, the small border city of Eagle Pass, Texas, has provided the backdrop for a bitter legal battle between Gov. Greg Abbott and the Biden administration over how best to handle record numbers of migrants arriving at the border. The court fights, which intensified this week, have centered on claims that the border is in crisis.

But recently, the opposite has been happening along the Rio Grande as it curves its way through Eagle Pass: In an area that last year was the epicenter of unauthorized migration along the southern border, far fewer migrants have been crossing.

Mr. Abbott has cited the slowdown as evidence that his aggressive attempt to push the boundaries of immigration law and his $10 billion program to harden the state’s border with Mexico — using National Guard troops, razor wire, helicopters, boats and floating buoys in the Rio Grande — has been working.

“The cartels have rerouted their routes to cross the border because Texas is the only state that’s putting up any resistance,” Mr. Abbott said during a news conference in Eagle Pass last month, flanked by more than a dozen Republican governors.

If the federal government did what Texas is doing, Mr. Abbott added, “you would eliminate illegal immigration overnight.”