Orkine, don't use the crane method, it's dangerous if you don't know what you are doing.
I have personally moved/transplanted 6-10 adult size trees and can give you some advice from my experience (50% success).
1. Forget moving a large avocado tree, not worth the effort, their roots are long and will not take the transplant.
2. Mangos that are large, 6" diameter trunk or more is difficult to move unless you have machinery to dig deep. They seem to have longer tap roots. I have not had any survive the transplant.
3. Link from Stephen by Truly Tropical is the best method of removing/transplant fruit trees or any other large trees. Key is to trim back branches as much as you can.
4. Best to get professional help, they have the trench diggers and equipment to move the heavy tree with rootball, you will have a hard time moving the heavy tree with soil attached.
-trees I have transplanted/move: Aprium (6" trunk), Citrus (3-5" trunk), Avocado (6" trunk), Mango (5" trunk), Plumeria (5" trunk), loquats (6" trunk), pine trees, redwood trees by myself and its a lot of hard work.