The Imperial Russian army and succeeding Reds with Rockets during the Soviet era were this. Although media tends to only focus on the ethnic Russians since they were dominant, there were lots of Armenians, Georgians, Turkic, Siberian, other East Slavic, etc. forces within the military. It's probably one of the best examples of this trope, and all the more ironic when the Nazis were defeated by an army of people they considered ethnically inferior.
The New Russia is a downplayed example, as Russia is now one country, but still as a very diverse population in its army.
The Warsaw Pact was made up of various Eastern European nations, which is nothing to say of the Soviet's various Asian, Latino, African and Middle Eastern allies all united by their opposition to NATO.
And ironically, many ex Warsaw Pact nations have joined NATO, which itself is a good example of this trope. NATO comprises of numerous North Atlantic nations.
Many of the armies of large empires of world history were this, for instance Xerxes' army and navy that invaded Greece during the second Graeco-Persian war included contingents from pretty much every nation from Egypt to the Black Sea, from Asia Minor to the Indian marches.
The British Empire forces in World War I. The United Kingdom of Great Britain and Ireland, Australia, Canada, Newfoundland, New Zealand, India and East, West and South Africa.
Bill Slim's Fourteenth Army in World War II, "a huge foreign legion of what Attlee called 'the scrapings of the barrel' from half the nations under the sun," as George MacDonald Fraser put it. At the time, India was not a unified country, but a conglomerate of different princely states and tribes - Sikhs, Baluchis, Jats, Rajputs, Punjabis, Dogras, and many others, all of which had their own regiments. The Indian Army also included units from all four countries of the UK, Gurkhas and Afghans. Fourteenth Army drew primarily from the Indian Army, but also included Burmese, more troops from Nepal, Anglo-Indians, two West African divisions with troops from Nigeria, Gambia, the Gold Coast and Sierra Leone, and an East African division with troops from Kenya, Tanganyika, Nyasaland, Uganda, and Rhodesia. Not to mention significant external support from Nationalist China and the United States, which provided most of the air power and a Chindit-trained infantry regiment, Merrill's Marauders.
The Axis, specifically Nazi Germany itself, had an Evil Counterpart in -of all things- the Waffen-SS foreign volunteer divisions. At first joining was limited to ethnicities classified as "Aryan" and thus "worthy" in the racist ideology of the Nazis, such as Scandinavians (Panzerdivision Wiking), Dutch, Belgians and French (Panzergrenadierdivision Nordland and others), organized in several divisions. Later on, recruitment was gradually opened to volunteers from other Axis countries - Hungarians, Romanians, Italians, as well as "ideological brethren" regarding anti-Semitism such as extremist Muslims from the Balkans (mountain division Handschar). And finally, when the Allies started gaining the upper hand and German casualties were mounting, it was, paradoxically, the Waffen-SS (the very embodiment of the regime's murderous ideology itself) that opened recruitment to just about anyone who could fight and, for some reason, wanted to fight for the Germans, including people classified as "racially inferior" - Western and Eastern Slavs (a lot of Russians and Ukrainians, specifically), Central Asian peoples, Georgians etc, turning the Waffen-SS into quite a multinational force (which was exploited by official propaganda presenting it as a "European defense against Judeobolshevism"). However, Equal-Opportunity Evil certainly did not apply, neither in theory nor in practice, as the entire officer corps was predominantly German and the foreign volunteers were nearly exclusively enlisted -seen as cheap cannon fodder-, and as there was certainly no German plan of multinational pan-European unity after the Endsieg, but one of German hegemony.
The British Eighth Army had British, Scottish, South African, Australian, Indian and New Zealand divisions, and smaller units of Free French, Czechs, Greeks and Poles.
Another WW2-era British outfit, and one of the iconic commando units, no less: the No 10 (Inter-Allied) Commando, which included Frenchmen, Dutch, Belgians, Norwegians, Poles and even a German troop composed either of German Jews or (to a lesser degree) non-Jewish German political refugees.
The modern British armed forces count as well, come to think of it; recruitment is open to citizens of all Commonwealth countries as well as Ireland, and EU citizens or other immigrants can join if they've been resident in the UK for seven years.
The Austrian/Austro-Hungarian Empire was very much this trope. Although there was a predominant German (and Hungarian) veneer, the Habsburg domains were home to various peoples, cultures and beliefs, all more or less united by their loyalty to the clan. By World War I, its military forces spoke multiple languages that part of an officer's job was to learn most if not all of them. Though at the time, this also gave the impression that, compared to Imperial Germany, Austrians weren't exactly "pure" Germans.
Even more pronounced in the case of military alliances composed of several independent nations.
In the Seven Years' War the Prussian army and its allies (Great Britain and a few small German states) faced the armies of Austria, France, Russia, Sweden, Saxony, several smaller German states and of the Holy Roman Empire. That last one was in itself a microcosm of contingents supplied by the middling-to-minute principalities etc. of the Empire, each of which was a sovereign nation at the time.
The Allied and Associated Powers of World War One - France, Serbia, Russia, the British Empire, Belgium, Portugal, Japan, Italy, Romania, the United States etc. etc.
The 1991 Gulf War coalition, besides the US, Saudi Arabia and the UK, included over 20 other countries. Likewise, the 2003-2009 US/British-led Multinational Force in Iraq included around 40 countries during it's existence.
Although many of the armies from the Renaissance to at least the Thirty Years War were composed of soldiers of fortune from all over Europe in a similar way.
The planned Multilateral Force. Subs and Warships manned by multinational crews and equipped with nukes.
Many space missions, especially after the Cold War ended. Personnel from dozens of different countries have crewed Russian and American vessels side by side, not to mention the ISS, which is owned and operated by fifteen nations, and also the European Space Agency (18 nations). Major future missions, such as human travel to Mars, are likely to be international as they would be so expensive, and the need for Cold War one-upmanship has vanished.
Basically the point of Up With People. It's common to see upwards of fifteen different countries represented in a single cast. They also double as True Companions for the cast itself.
ESL programs often contain English teachers of various countries. More often than not, they are from countries in which English is an official language along with whatever host country has brought them. Japan, South Korea, Vietnam, and other countries have such programs.
Many modern charity organisations such as Amnesty International and the Red Cross count as this, being huge teams of like-minded idealists from across the globe.
Professional sports teams can end up this way, especially in high-profile leagues whose sports have wide international following (notable examples include the English Premier League, the National Basketball Association, and the National Hockey League). Such teams are often the most well-known and richest in their leagues, as they can find and attract both overseas talent and fans.
The Thirty Years War was full of this. Basically every army throughout the whole war was mostly mercenaries of pretty much every variety of Christianity and from pretty much any place in Europe. Ironic considering that the war was justified as being about religion and at the end most of Europe had been involved in some way at some point (most of them indirectly) and thus pretty much every soldier in the war had at some point fought against "his country", "his religion" or both. Wallenstein, one of the best known generals of this (or indeed any) war fought for the Catholic side yet was a minor Bohemian Protestant nobleman before it all began. That neither kept him from going where money and power were, nor the emperor from ultimately "arranging" for his violent death, when Wallenstein had outlived his usefulness and was getting too powerful.
The 2016 summer Olympics featured the debut of Team Refugees, for athletes displaced or expelled from their countries of origin. Selected by the IOC Executive Board, they received a standing ovation from the crowd at Rio's Opening Ceremony, competing under the Olympic flag.
Interpol was established in 1923 as the International Criminal Police Commission. Its membership spans 190 countries as of 2015, and works to coordinate member nations' law enforcement operations against terrorism, crimes against humanity, trafficking of drugs, weapons & humans, piracy, money laundering, computer crime, and other illegal activity spanning multiple nations.
In a non-military real-life example, a large percentage, if not the majority, of crews on civilian ships (merchantmen) are this trope nowadays. While on naval (i.e. military) vessels the crew typically belongs to one nationality (that of the state in whose navy the warship serves), on civilian, privately owned ships, there is no such restriction, and crews are hired from a large international pool of seafarers (many of them hailing from Asian and Pacific nations). Also, besides the Multinational Team, a peculiarity of modern merchant shipping is the fact that ships can be -and are- registered in any country: not necessarily that of the owner, but rather one with the most advantageous legal regulations for and lowest taxation on shipping. All of that leads to the funny -but common- situation of a ship being owned by a company or private person from one country, sailing under the flag of a second, unrelated country, and being crewed by people from a third country (or, more likely, from a third, fourth, fifth, sixth...as said above) again totally unrelated to the first two.
In an airline disaster, it isn't uncommon to have the investigation team be multinational. It's entirely possible the lead investigators will be from the country the plane crashed in, but other members include representatives of the country the airline is based in, people from a third country that manufactured the plane, experts brought in from a respected agency from a fourth country like the American National Transportation Safety Board (if they aren't already there), and as modern airliners are often assembled from assemblies made from countries other than the one it was finally assembled in, you can have people from a fifth, sixth, and seventh+ countries there as well. And if the incident is believed to be intentional, you'll have investigators from all the countries represented by the passengers and crew as well.
In disasters, search and rescues and other such situations, especially at sea, it isn't uncommon to see the flags of many countries involved. The Chilean Mine Disaster and rescue provides a textbook example where multiple countries provided equipment, personnel, technical expertise and fabrication in order to rescue the trapped miners.