I'm not sure of the details of how, but I believe the UN sent troops to fight in the Korean War. I also believe that was the first time we fought a war, refusing to call it an actual war and willing to accept something other than victory. That's pretty much what we've done since.
Yes as the US was forced into a toe hold in the Pusan Perimeter. The rescue force, mainly US was organized as a UN operation.
As we kicked the crap out of NK more countries joined in.
The eagerness to help/support faded when the Chinese came in on the side of NK.
I think the lack of victory somewhat can be blamed on our multi-national coalition method of war and also the Military Industrial Political Complex enjoying the contracts and kick backs that come from prolonged continual war.