Well, I don't know what was triggering the problem, but resetting the website - by adding a bogus key/value pair to the configuration, and saving it - triggers the website(s) to pick up the changes. Apparently the underlying issue is that the Kudu deploy doesn't seem to be triggering the website to reset itself. I'll add more details in the future if I run into the problem again.
[Edit 2013-10-15 - Today, deploys seem to be working normally again. My guess is that it was some sort of transient Azure bug that's now fixed.]