I think I would rather just take one of my bikes with me and pay the shipping. They are set up how I like them, and I know all of their quirks. I've also rode enough cheap bikes to not want to ride any of them very often.
That being said, I've seen a lot of interesting experiments out there. It would be neat to begin your tour somewhere with a cheap bike, visit various places along the way to upgrade the bike (thrift stores, craigslist, etc) by trading/haggling/word of mouth, and see what you could end up with.
Kind of like the paperclip experiment that somebody did. The guy started with a paperclip and posted ads to trade it for something slightly better. I think he ended up trading up into a house from a single paperclip. It would be fun to see if someone could trade your average beat up box store bike and end up with some nice bike like a Trek 520.