The 13 federal appeals courts across the country sit just below the U.S. Supreme Court. In theory, the appeals courts, each covering a judicial circuit, are equals.

But often enough, one circuit court has taken on an outsize role in shaping seminal constitutional questions that end up before the Supreme Court.

After Sept. 11, it was the U.S. Court of Appeals for the Fourth Circuit in Richmond, Va., which took a sympathetic view of the expansive executive powers asserted by President George W. Bush in the war on terror. During the presidency of Donald J. Trump, the U.S. Court of Appeals for the Ninth Circuit in San Francisco emerged as a check on some of the administration’s most contentious policies on the environment and immigration.

Now, the spotlight is on the U.S. Court of Appeals for the Fifth Circuit, as it weighs the fate of a Texas law that would give police officers in Texas the authority to arrest people suspected of having entered the United States unlawfully — a power that has long been understood to rest with the federal government.

It is a case with far-reaching legal and political implications, and it has thrust the Fifth Circuit into the middle of a fierce debate over the extent of federal power over the nation’s borders.

Based in New Orleans, the Fifth Circuit hears cases from three states — Texas, Mississippi and Louisiana. Twelve of its 17 judges were nominated by Republican presidents; six of those 12 were nominated by Mr. Trump.