Movers don't take plants, animals, hazardous materials and a whole list of other chemicals, fuel, ammunition, fireworks or any other explosives, but they will pack your garbage and take that for ya
Anyway, having just moved way too many houseplants, including the housetrees, shrubs, and vines, I would add that a roll of shrink wrap comes in handy to coral the top growth into a loose sleeve of sorts, keeps them from getting tangled in each other and breaking branches/leaves. Stack-able crates work great for the smaller stuff, they breath and you can bungee them together and in place, helps to pack tight or inter-pack with something for shifting, layer where you can and top-pack with something breathable and compressible yet firm for the bouncing (like shredded crinkle paper or Excelsior or the like).
My honey came up with an ingenious method to transport the hanging plants hanging

so we didn't have to disturb them too much, but it wasn't in the back of an SUV.