Sunshine (2007). The crew of Icarus II are American, Chinese and Japanese, as they were the countries most likely to have major space programs when the movie is set (in 2057). India and Brazil were also suggested, but it was decided to leave them out to avoid a too-disparate cast.
The Transformers Film Series has a partial example in NEST. While only American and British forces have any serious screentime in the film representing humanity alongside the Autobots, the prequel comics suggest the involvement of other nations, while the prequel novel features NEST agents from Israel, Russia and Japan. Michael Bay wanted Bundeswehr troops, but this was vetoed by the German government.
A classic example would be Sahara. Here we had an American tank crew (commanded by Humphrey Bogart) pick up: a bunch of British Soldiers (one upper-crust officer, one working class), one Aussie, one South African, one Sudanese (British Colonial), one Free Frenchman, and two prisoners (one very Nazi German pilot, one harmless Italian). These (Western) Allies in miniature then hole up at the only water source for 100 miles and try to play Alamo with a German battalion. Though in those days, it was more of a propaganda emphasis on Allied unity in the face of the Nazi threat.
Mystery Science Theater 3000 once had a movie called 12 To The Moon: It featured a Nigerian Muslim pilot (who exclaims "Praise Allah!" when landing the ship), as well as scientists from the USSR, France, Japan, and Turkey. Not to mention the Jewish guy who discovers his German comrade is the son of a notorious Nazi (they eventually become friends just in time to make a Heroic Sacrifice together). Of course, the mission is led by the hunky all-American beefcake guy, a fact that Mike and the 'bots are quick to lampoon. Still, considering this movie was made in 1954, it's actually a legitimately impressive stab at diversity, despite not being completely free from Unfortunate Implications.
In Pacific Rim, with the entire world in danger of being attacked by Kaiju, the Pan Pacific Defense Corps is formed by 21 nations across the Pacific Ocean to contain, control and eliminate the Kaiju. The four remaining active Jaegers are made from and piloted by people from various nations, including the United States, Japan, Russia, China, and Australia. France has also provided some of the science and Great Britain has provided pilots. Several of the top scientists are German or Chinese.
Elysium: Delacourt is French and the president of Elysium is Indian. Only three characters in Los Angeles are Caucasian (the White Male Lead [though with the name DaCosta and his espaņol-hablando childhood, he could be a light-skinned hispanic], his unnamed supervisor and Carlyle), and there's a token black dude, but the rest are Latino, and Max is fluent in Spanish. The only East Asian we see is an Elysium biotechnician in one scene.
The Cleaners shown in Underworld Evolution (and described in more detail in the novelization) are made up of soldiers from all over the world, including special forces and intelligence agencies (CIA, MI6, and Mossad are mentioned).