Answer: This is a tough question, and I'm not finding much help in my references. In fact, one reference says the trees are protected, and it's illegal to remove them. I'd check with your state's Dept. of Natural Resources before doing anything. They may also be able to help you.
If possible, wait to transplant until fall, so they have the winter and spring to develop root systems before having to fact the heat of summer.
Here are a few web sites that might help. The first one is tips for planting Joshua trees; the second is a fascinating account of the history and biology of the plant.
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