I was surprised to see specific replies to this question, since it would be done significantly differently on different types of BUDs. I've owned 4 of them, and they were all different.
On some of them, the easiest way would be to take apart the various sections, but other dishes don't have separate sections.
On some dishes the actual dish can be taken off separate from the mount. The one I'm using now just has 2 bolts, and the dish comes off the mount.
On other dishes, you'd pretty much have to take the whole mount off the pole with the dish attached.
But every dish has it's own quirks.
One thing though, particularly if you have to remove the dish and mount as one unit, unless you have multiple people to help, and can get all these people up high enough to be useful (once we backed up 2 trucks and a trailer to the dish, and had people standing on each, all helping to lift the thing), one thing that can help to make it a one person job is to set up some form of a gin pole. On one very heavy H-H dish that I took down by myself, I clamped a secondary pipe or 4x4 or something (I can't remember what exactly) to the side of the pipe, so that it extended about a foot above the top of the dish's mount and had a pulley at the top of this secondary mast. I ran a metal cable through the pully, and attached the short end to the mount, and the long end to the tow hook on my car. I then took up the slack by moving the car. I then went up on a ladder, and lifted the whole dish/mount off the pole, lowering it a few inches until the metal cable supported it. Of course, the dish flopped down a bit so one side was touching the ground, but at least the heavy mount was supported. Then, I just slowly moved the car towards the dish, and the dish slowly sort of rolled around the pole a bit as it lowered toward the ground. If I could have found even one person to help, it would have been a very smooth operation, but even so, it worked pretty well. On the other hand, on another dish, I wasn't so careful, and the whole thing came crashing down (actually this was while putting it up). So it does seem worthwhile to have some sort of support there in case you lose control of the thing.