The first time through this thread, I forgot the story of a coworker who lost his ID.

He lived in Conecticut, but he was on an extended assignment in Wisconsin. He lost his CT license and applied for a replacement by mail/phone.

He realized that he was supposed to fly someplace that weekend and tried to get a WI ID Card so that he wouldn't have trouble.

He was caught immediately and almost arrested. He was detained for some time while they went through his story, but apparently they immediately had access to the fact that he had applied for, but not been issued, his CT ID a few days earlier.

So, I have no idea what you should do if you lose your ID while travelling, but here's an example of what not to do.