Go to the view where you are looking at thumbnails of all your documents. Tap "edit" in the top right corner. Select document(s) you want to copy to WebDav. Tap arrow icon in top left corner. You'll now see several options for sharing the file(s), one of which is copy to WebDav.
Cool. Thanks for the instructions. I've just tried it, and it's not very intuitive but it'll do
I initially thought that iOS Pages would save incremental updates to the file, say every five minutes. But this would create a lot of network traffic.
I'm actually quite happy to leave the doc on a WebDAV share and, if I edit it using Pages on an iPad, simply reshare with the WebDAV once I've finished.
I'm surprised how seamlessly WebDAV shares work on OS X on my Macbook. I can sleep the notebook and when I resume the WebDAV share reconnects as if nothing has happened. The only issue is if I go to a new location with a different IP address while the notebook is sleeping. I get a confused error message.
Proper invisible syncing would be better. I'm really hoping this arrives with iCloud's integration into Mountain Lion, and that Apple doesn't keep-up the weird segregation of iOS files. I've not really seen anything like it yet in Mountain Lion, however (I'm a paid-up dev).