The people at Greer Gardens, who specialize in rhododendrons and azaleas, have told me that these plants can be moved at any time of the year, as long as the ground isn't frozen. It's their shallow root systems that make that possible. The root ball is not very large. I moved a 7 ft tall azalea a few years ago -- in July in fact -- and it's doing very well.
The trick is to give that shallow root system water 3 or 4 times a day after it's replanted. Just a good sprinkle with the garden hose seemed to be all my azalea needed, no deep watering or anything like that. The water table in my yard is quite high, even in the summer, so if your soil gets dry deeper down, your plant may need a deep watering once a week with a light sprinkling several times a day in between. The roots need to be kept moist, not soggy wet, but you can't let them dry out, either.
The other thing is to dig up as much of the root ball as you can. The plant I moved had a root ball about 3 1/2 or 4 ft in diameter and about 2 ft deep. I couldn't move it myself, but the guys I had working for me were able to. I had them put it on a tarp, so it could be gently dragged to the new planting hole, which we had already prepared. As long as you're careful when moving the plant, so you don't damage the roots, everything should go fine.
I don't remember pruning the plant back, but it wouldn't do any great harm, as long as it's finished blooming. The time to prune these plants is right after they have finished flowering. If you wait much longer, you won't have any flowers the next year, because this is the time they set buds.
The guy at Greer Gardens showed me a 12 ft tall rhodie they had just brought back from a show. It was beautiful. He said they routinely dig them up, take them to shows, bring them back and replant them.
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams