I have moved bikes in box-style moving trucks many times, but I've never moved more than two at a time. Like DoctoRot, I started with a wheel chock from Harbor Freight.
www.harborfreight.com/motorcycle-wheel-chock-69026.html
But I made a bike stand in the shape of a "T" using two pieces of 2x10. The top/horizontal part of the T is about 5 feet long, and the vertical is the length of the bike. The vertical lays over the top of the horizontal, screwed together with deck screws. The chock bolts to the vertical (and since it's raised up the bolts don't hit the floor), and the ends of the horizontal have U-Bolts with flush nuts on the bottom. The bike rolls into the wheel chock, and I run ratchet straps from the handle bars to the u-bolts. Now the bike is locked to the stand and can't go anywhere. It's very stable. I'll try to get a picture tomorrow.
All of the wood and hardware came from Home Depot. Actually, the last time I did this I used a one-way truck rental. I brought just the hardware, screws, and a drill/screw-gun. I went to a local Home Depot, had them cut the 2x10 into two pieces, and built the stand as I picked up the bike.
Of the trucks I've rented, the Penske trucks have metal floors, and the U-Haul trucks have had plywood decking. I have never seen a box style moving van with D-rings or tie downs on the floor of the truck. I've only seen them with wood rails along the walls.
With the wood floors, I used 3 or 4 deck screws and just screwed my stand to the floor. When I pulled them out I couldn't see the holes. This is how I moved 2 bikes from Boston to Austin. With the metal floors, I've used straps to hold the stand to the rails on the walls of the truck. This worked fine with only one bike, with the stand pushed up against the front wall. I'm not sure how it would work with 6.
The other issue is getting the bikes into the truck. I think as the trucks get bigger, the deck height gets higher. With 3 strong folks, or 2 strong young folks, you
can get a big-four KZ up the ramp of a moving truck, but it's not fun. The last bike I moved was a GL1500 Goldwing, and we used a scissor lift built into the floor to get it in. That was really nice, but getting it out took 5 people. I wasn't brave enough to just ride down the ramp.
Even if you found a lift-gate truck, the lift gates I've seen are wide but not deep. So the bike would have to go on the lift facing sideways, and you couldn't roll it into the truck. The movers that did my bikes had a lift gate, and once they brought it up to deck height they still had to basically pick up the bike to turn it into the truck. I suppose if you had a dolly stand like this:
loadall.com/products/loadall-garage-dolly
It's true that the standard in the professional moving and freight industry is to strap the bike to the wall of the trailer. I tried to watch them do this to 3 of my bikes, but eventually I just had to leave. I couldn't take it. That is NOT the way to move a motorcycle. Sure they were using moving blankets, but they strapped right across the gas tank! They put a wave in both my Z1R tanks, and bent the exhaust hanger/rear peg mount on my GPZ1100. Never again. There are movers that specialize in bikes. They have special pallets similar to what Nerdy described. I have no idea where you might get one. Google didn't turn anything up.
The best experience I had was a U-haul single motorcycle trailer. The low height and wide ramp made it easy to roll on, and the trailer was set up with wheel chock and tie downs and in the right place. I pulled it with my Integra (a fancy Civic), so it was a light trailer. If there is any way you could pull a trailer, that seems like it might be the best route. But it would have to be a big one.