I would start with a lot of new water. Use 50-100 gallons to wash live rock and sand and keep 50 gallons of old water. I never keep more than 35% of my old stinky water. If your water is good quality then they will appreciate a good water change. If its bad its going to be hard to move successfully no matter what you do. Speed is the important thing, the faster you move the better off you are, no matter how you do it. Nothing kills fish deader than leaving them in buckets overnight.

I always get a couple coolers and 6-10 5 gallon buckets. The live rock and sand go in the buckets, no water, (and less chance of going anaerobic.) And all the fish and corals go in the coolers 1/3 full of water. The coolers keep the temp stabilized too. I would also take 6-10 5 gallon buckets with clean salt water from the surface before you start the teardown. That way it won't be filled with sediment. The rest of the water can be used to clean filthy old rock and sand. There's no point in moving a dirty sandbed(DSB?). That's just asking for it to die. And a true DSB is an even worse idea to move intact. They take a year to get going and an hour to kill in winter temperatures.

30 gallon trash cans from home depot/lowe's are <$10 and quite handy, but extremely heavy. A pump to move the water back and forth is mandatory. I recommend a python with a eheim 1260 & a 3x3/4" nipple to adapt to the python.