TeamViewer for the iPhone and TeamViewer HD for the iPad have received new updates that provide support for transferring any file from an iOS mobile device to a computer (and vice versa). Windows PCs as well as Mac OS X Lion and Linux systems are supported.
TeamViewer is free for private use, and will allow transfer pictures, videos or documents over a wireless or 3G network.
I tried a couple of file transfers using my iPad and a remote Mac and all worked as advertised, although the file transfer screens only worked in portrait mode, which seems an odd oversight. Some other remote access apps have file transfer features as well. LogMeIn Ignition comes to mind.
Apple iPad Air 2
Apple iPhone 6s