Mr. Trump’s order, which suspends entry of all refugees to the United States for 120 days, created a legal limbo for individuals on the way to the United States and panic for families who were awaiting their arrival.

Mr. Trump’s order also stops the admission of refugees from Syria indefinitely, and it bars entry into the United States for 90 days from seven predominantly Muslim countries linked to concerns about terrorism. Those countries are Iraq, Syria, Iran, Sudan, Libya, Somalia and Yemen.

It was unclear how many refugees and immigrants were being held nationwide in the aftermath of the executive order. The complaints were filed by a prominent group including the American Civil Liberties Union, the International Refugee Assistance Project at the Urban Justice Center, the National Immigration Law Center, Yale Law School’s Jerome N. Frank Legal Services Organization and the firm Kilpatrick Townsend & Stockton.

The lawyers said that one of the Iraqis detained at Kennedy Airport, Hameed Khalid Darweesh, had worked on behalf of the U.S. government in Iraq for 10 years. The other, Haider Sameer Abdulkhaleq Alshawi, was coming to the United States to join his wife, who had worked for a U.S. contractor, and young son, the lawyers said. They said both men were detained at the airport Friday night after arriving on separate flights.