We moved a lychee tree that had been in the ground for 14-15 years. As some comments have alluded to, it's a massive job with lots of heavy lifting and digging. We did it all by hands. The tree had to be moved to clear the way for a home renovation project. At first, I tried to spare some branches. But in the end, I had to trim all of them, sparing only the main tree trunk, to lessen the weight and to make handling easier. The tree amassed a hefty trunk and root ball, even after pruning. I tried to rig up some contraption to get it out of a 4 foot hole. In the end, it was human ingenuity and sheer sweat power. We simply backfilled part of the hole until the tree rose out of the deep hole. The job took place in January when the weather in California is cooler. There was not much rain that year. I put the tree into a new hole and hoped for the best. About 1 month later, new shoots started coming out. In the end, the tree survived and prospered in its new location for the last 2.5 years. This year, it is blooming up a storm.
I also agree that you first want to decide if the tree bares tasty fruits before tackling the move. It's definitely doable.