I moved a bunch of mounts 1500 miles from CO to BC in my cargo trailer and pickup canopy…multiple loads. Size ranged from pronghorn up to moose including elk and kudu. We used a pile of packing blankets, soft furniture, lots of cord and rope to stabilize them etc. Really did not have any issues. Just do not let antlers dig into capes etc. Some were upside down resting on the antler tops and top of shoulder, others on the back of the mount, etc.
About the only visible damage was a small spot an a 35 yr old mule deer mount with very brittle hair.
For a moving company to crate them would have cost $500 just for each mount box plus the cost of shipping was quoted extremely high due to extreme volume and weight. Not practical or cost effective for us.
Was really nice to get them up in our house build! But that took having scaffolding, etc. Not easy, not to mention paperwork at the border to bring them to Canada as part of our move up here.
Good luck getting you mounts moved.