It differed from nation to nation, but usually they largely sat around and did nothing. They had ill-defined objectives, and were largely there to secure resources/arms/wealth that the Allies had sent to the Tsar with an unrealistic mandate to help anti-Communist efforts in the background looming over them. The British in particular supported Kolchak's Siberian Government, but their aid to them never became especially helpful and was too limited to achieve their own objectives.
As the situation turned around for the Whites, they quietly left one after another, having not done much anyway despite some small-scale clashes and largely achieving their objectives in port/border cities they became situated in. The Japanese were probably the most ambitious and tried to make a puppet state out of Vladivostok, but it too fell apart rather quickly, especially when the Japanese interests shifted once Bolsheviks gained domination over Mongolia.
The most important foreign power to become involved in the Russian Civil War was probably Poland, which over the course of the Polish-Soviet war was able to retain large parts of Ukraine and the Baltic's for itself as well as prevent a general Soviet expansion into greater Eastern Europe.
In any regards Soviet history likes to play up the foreign intervention, and it did damn the White Movement in many ways in the eyes of common Russians. However it was not militarily decisive in the conflict, and totally uncoordinated.
I can't provide a broad answer, but I can point you to some sources regarding the US Expeditions. These included the 31st Regiment ("Polar Bears") and 27th regiment ("Wolfhounds"), who I believe were both garrisoned in the Phillipines and deployed to Siberia from there in 1918.
http://en.wikipedia.org/wiki/American_Expeditionary_Force_Siberia
Additionally, the US 339th Infantry Regiment (also "Polar Bears") and supporting units deployed to Arcangel in 1918. The main mission of both expeditions was to safeguard war material from the Bolsheviks, and the expedition to European Russia also engaged with the Bolsheviks, taking heavier casualties.
http://en.wikipedia.org/wiki/Polar_Bear_Expedition
The University of Michigan maintains a physical and digital archive of materials related to the Polar Bear Expedition here: http://quod.lib.umich.edu/p/polaread/, and the "Detroit's Own" Polar Bear Memorial Association has an archive of primary sources, including some diary entries, as well as later recollections by family members, here: http://pbma.grobbel.org/polarbearstories.htm.