Yes, it is probably to do with your diaphragm, but there are a load of muscles that also go through the diaphragm, and if they are tight/spasming/whatever, they may well be causing the diaphragm to spasm as well, causing the stitch.
The fact that it hurts on the downhill would kind of vaguely indicate that it hurts when you do longer strides (most people tend to stride longer when running downhill), and the main muscle that is involved in that motion, which elongates more with a longer stride is the psoas.
Helpfully, the psoas attaches pretty well right up into where the diaphragm also attaches, so its a relatively good guess.
Here are a couple of blogs I wrote about it. The first one is about the issue in more detail, and the second one gives a couple of stretches which should/might help, depending on if it actually is the psoas which is the issue. (there are another couple of muscles which can cause similar pain)
http://globaltherapies.wordpress.com...while-running/
http://globaltherapies.wordpress.com...nd-prevention/
If you are still having issues and you are anywhere near the Peak district, drop us a line.