Here is what I would do:
Buy a styrofoam cooler for each one of your tanks. Double line it with thick trash bags. Pour some of your old saltwater in the cooler and add a small bag of carbon - the carbon will soak up some of the toxins from those mushrooms being so stressed. In fact, I would try to bag mushrooms separately. Then, check your local weather and the weather at the locations that you are going to drive through. If the temps are going to average higher than 83F, tape a couple cool packs to the lid of the cooler and have a couple more chemical cool packs on hand to replace them when they are spent. If your temps are lower than 75F where you are traveling through, tape a couple of hand warmers to the top. If you seal off the cooler completely with tape and you have that handwarmer in there, it will use all of the oxygen inside the box - so don't do this unless you have your corals and everything packed tight in bags.
There are two ways you could transport... in individual bags, or in a semi-open container. I personally would put everything in bags and make sure the temp is pretty constant by the heat/cool packs and being in a styrofoam bag. IF you do the open container, it would be very wise to add a battery powered air pump for each container and have extra batteries on hand. But, if you do open container, do not seal in hand warmers and make sure to pack your mushrooms separately.
That's pretty much it... it takes a little luck, but 36 hours really isn't that bad. You will have to consider setup time on the new systems as well... Make sure you save any of your old filters and keep them wet in old tank water for the trip. If you keep it wet, you can quick cycle the new tank. Try to start the new tanks with completely new dry aragonite - you won't introduce old nitrates and phosphates into the new system. Keep a cup of your old live sand for each tank and keep it wet for the trip.
Well, now that should be pretty much it. lol
