They use DHL, there was an article in the Singapore papers last week and they interviewed the guy who runs the show. He's been running if for a long time now and said he never lost a single item before. He also said one of the most challenging aspect is when they need to fly in the parts for some last minute repairs.
But funny that they don't use their own planes, uses Singapore Airlines instead to ship the stuff from Singapore to Japan...Not sure if they always use the national carrier of the last host country.
I worked with DHL before, and one key thing about DHL Express, who is in charge of F1 logistics, are their expertise in negotiating with customs, and have a great rapport with customs all over the world in case they need to ship in or ship out urgent items.
Remember though, under the DHL umbrella, there's DHL Mail, DHL Supply Chain, DHL Global, and also DHL Express. It's really tough to get a job in DHL Express.
They do make it sound big by not losing a single item, but trust me, in DHL standards, it's a piece of cake ;)