You need to realize that the Red Cross isn't one huge organization, it's actually a group of organizations that share funding , organization, methods, and ideals. This is because local governments tend to react better when the people trying to help are also locals who speak the same language and know the same customs.
Hell, some of the organizations work under a Red Crescent instead of a cross because of their Muslim leanings and a Red Crystal is used in places where nether option seems appropriate (like Israel, who tried to get a Red Star of David for a long time).