Union Pacific and BNSF, the affected carriers, said automotive, agricultural, chemicals, and other consumer goods were halted.
The Justice Department has asked a federal court to order Texas to remove the buoys, saying the water barrier poses humanitarian and environmental concerns.
When American forces withdrew from Afghanistan, Abdul Wasi Safi went into hiding and learned that friends in the Afghan military had been killed by the Taliban.
Under the new system, migrants apply directly to the agency and a government official will determine who gets in.
Chief Justice Roberts granted a stay pending further order and asked the government to respond by 5 p.m. Tuesday.