What's your question? Ask

Did the ones without leaves ever have leaves? If not, I would say they may be dead. You can give them the scratch test to see. Burning bushes are moderate growers and you should have seen some growth by now.
Check the stems for signs of scale or fungal issues.

You may need to apply Neem oil to combat both. Also check their cultural needs in the article below to be sure they are getting enough sun and water.

https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm

California spans USDA zones 5-11. This means that burning bush will grow in every part of California. The question becomes whether the area that you are moving is in the upper portion of that range. If this dips into zones 8-10, the shrub will become highly invasive, and introducing something like this to this particular region will be very irresponsible and potentially illegal depending on local regulations.

To answer your question, yes. It will tolerate the move. Sometimes a little too well.

Here is an article that will help you to know more about these shrubs:

https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm

That last sentence sums it up quite well.

These have been overwatered. This can be very hard to know since it is covered in stones. This will hide the soil so that you cannot see how dry it is, and can hold it more moisture. I would recommend using actual mulch, or even none at all with these shrubs. They are quite drought tolerant, but will succumb to disease, easily.

At this time, you will need a fungicide. Biofungicides can help keep the soil from getting too infected, as well.

Here are some articles that will help:

https://www.gardeningknowhow.com/garden-how-to/info/using-fungicides-in-garden.htm

https://www.gardeningknowhow.com/special/organic/biofungicide-information.htm

https://www.gardeningknowhow.com/garden-how-to/mulch/benefits-of-using-mulch.htm

https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm

Burning bush responds well to trimming, even severely cutting it back to within 3 inches of the ground to rejuvenate the shrub. It's best to wait till late winter or early spring to prune, however.

Here are more pruning tips:

https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/pruning-burning-bush.htm

Seeds from a dwarf or miniature would be a hybrid and therefore the seedling won't be true to the parent plant.

You can mix the seeds in a moist jar of sand with a lid or in a plastic ziploc bag. Keep them in the fridge for 3 months for stratification. They need the cold dormancy before they will sprout.

In the spring, plant them outside after frost or start them indoors.

The picture is quite blurry, so it is difficult to get a good look at the damage. Most trees and shrubs will need watering to be halted throughout the winter, and this looks to be damage from overwatering.

Pruning anything dead off will help, as long as it is completely dead. Doing too much pruning will result in the loss of the grafted top, however.

Here is an article that will offer information on the care of the shrub:

https://www.gardeningknowhow.com/ornamental/shrubs/burning-bush/burning-bush-care.htm