Airlines in Europe and the Middle East responded to the suspension of tightened U.S. immigration rules by again allowing passengers from countries that had been blocked to fly to the U.S., following chaos at airports and protests across the country.

A federal judge in Seattle on Friday issued a ruling that temporarily blocks President Donald Trump’s executive order on immigration and refugees signed on Jan. 27. With the directive, the president barred individuals from seven Muslim-majority countries from traveling to the U.S.,...