Yesterday a friend sent me an mp4 (video). I was on the computer that I also use for syncing my iPhone and iPads, so immediately stuck the thing into iTunes and then went over to my iPad, picked it up (it wasn't plugged into anything) hit Settings > General > iTunes WiFi Sync and clicked on "Sync." I had already previously set up the device to wirelessly sync at the time I was organizing things and had it synced to the computer via a USB cable. I was tickled to see the video shortly appearing in my iPad. Once one has the device configured for wireless sync it's easy to click on that sync button at any time and immediately sync new material. Now, this was because I had already installed the video into the machine that is where I do my syncing anyway and because I also had things set up to back up to the computer as opposed to backing up to iCloud. I don't know how things would have worked if I'd, for instance, been on the iPad when the friend sent me the thing in the first place, or on a different computer.....
We do have to be in the same network with the computer with which we're syncing and the computer has to be on, iTunes needs to be open. This is somewhat different than backing up to iCloud, which is a whole 'nuther thing...... And, right, when we purchase a video from iTunes we can expect it to download automatically into our computer(s) but not on an iPhone or iPad. I imagine that this is for reasons of concern about capacity and such. Last week I purchased a video through iTunes and it automatically went into my computers but I had then to manually sync my iPad in order to get it in there as well.