If you're really serious about making a movable box, here's what I'd do. I'd build the bottom out of 2x6 pressure treated lumber or maybe the 5/4 thickness, use 4x4s for cross braces below them, and use heavy duty casters (wheels) mounted to the 4x4s. I'd put weed cloth above the 2x6s to keep the mix from coming through, but the space between boards would allow good drainage. Dry, this thing won't weigh too much, but when the mix is wet, it's heavy, so don't skimp on the casters you pick. From experience with other projects if you do it with either be very hard to move/steer or a wheel may break. I'd guess you would need at least 300 lb. rated casters. Just a guess there though. You may also want to get locking casters so the wind can't move it around (or you cam make your own cheap locks that will work about as well. Good luck!

Using the term "all wood" to include plywood is the worst kind of uneducated snobbery designed to force consumers into "upgrading" unnecessarily. Partical board isn't any less "all wood" than plywood is! It's just smaller particals (thus the origination of the name) vs very thin sheets of veneers. In a flood situation, plywood is NOT any more resistant to moisture damage than partical board is. Some grades of plywood delaminate quite easily. It's all about the grade of each product and the glues that hold it together. The furniture board partical board plywood isn't the 16 pounds per square inch that those round "decorator tables" that you buy from WalMart are made of. That type of partical board DOES fall apart almost instantly when exposed to water. Furniture board partical board is made from much better glues and uses much higher pressures to set it. It's so much more "all wood" than plywood is that it is a LOT heavier than plywood! It wears just as well as plywood boxes if the proper construction techniques are used for each type of material. There really isn't a detectible difference to the end product at all once it's installed. BTW, if you want a matching interior, veneer actually installs and lays much better over MDF than plywood. And, veneer is "real wood" too! If you get matching interior on plywood cabinets, the top portion of the plywood is veneer also. Same materials. Same appearance. Same wear. "Upgrading" to plywood is money pulled out of your pocket and put into the pocket of the manufacturer and showroom. I NEVER suggest plywood to any of my customers. I always suggest putting that money into organizational conveniences that will make a difference how you live! That IS money worth spending!

Using the PT ply in your case is a good idea. I recommend screwing the PT ply to the floor joists with 2&1/2" deck screws) You can add glue i9f you wish. To add a finish layer, install(1/2" staples) a layer of 15 pound roofing felt---overlapping each row about 4" and install the finish ply---making sure to not place the new sheets exactly over the PT ply.(you will only need to make sure the first new sheet is different---each succeeding sheet will be different automatically) Then screw the new layer to the PT with 1&1/2" or 2" wood screws(deck screws so the PT ply will not rust them.)

I seem to have gotten confused on the Shenandoah construction thicknesses. For one, sides are 1/2". So disregard whatever I noted if you're considering this line. Funny, I'm looking over these nearly 40 year old junkers that are still standing. They were even uninstalled and shuffled around 30 years ago. The cheap wood grained laminate doors had been replaced. The sides and back seem to be a 1/4" particle board type product. Some of it also looks like luan but is painted and might be particle board. One shelf is loose and I can see the particles on the box side. Particle board 3/4" shelves chipping along the edge but strong. It seems to be solid 3/4" framing for the faces and hanging rail. They are still standing and doing their job! The side panel on the sink base has popped out and it's really bad construction otherwise. The boxes were painted and doors replaced so they don't look as bad as all that. The painted raised panel replacement doors have held up well for nearly 30 years. They look to be solid wood. The only real damage to the finish is on the raised panel edge of the two drawer fronts under the counter. They've lost finish and are chippy because the counter's drip edge doesn't project out far enough. I'm very surprised to see that situation on a lot of store display installations. The cabinets are just about even with the 25" counter edge. That's bad news for finishes. I'll be getting 26" counters for a good drip edge to protect the finish below. Right, the finish will probably not last 30 years and I'll have to deal with repainting whatever I get. I doubt the stains these days hold up that long either. A lot of them look layered and baked on like the paints to me. But hopefully the cabinets themselves will be decently built and still holding together well. I don't know why that would change unless someone was hanging on the doors or overloading areas with very heavy items, as livewire mentioned. The older builder type homes from 40-50 years or so ago seem to get there. They can just look dated at this point. Maybe they all do this but Shenandoah is calling their mdf type cabinets 'engineered wood' which I think is misleading since it makes it sound like standard construction is plywood.