I don't think it would be fair for us to try to provide you any sort of definitive answer. Some trucks bounce pretty badly even on good roads. But even a smooth-driving truck is going to lurch and jump if you have to park on a poorly maintained dirt lot while you wait for a load.
I would strongly suggest that you make two sets of plans for your pets, and decide after you actually get in a truck assigned to you. Plan 1 for if the truck has a good, smooth ride, and plan 2 for if it does not.
If you get stuck in a rough rider, then after a few weeks or a couple months, approach your company about a different truck, and see how that one handles.
Try to time your truck swaps for right before home time, so you can either go get your birds of you swap from a rough truck to a smooth one, or bring them home if you did have a smooth truck, but are swapped into a rough rider for whatever reason.
Yeah. Seems like that's what the bounce issue is going to boil down to.
But, the only way to know if there was a more definitive answer was to either wait and see, by which time it might be too late, or to ask.
So, I asked. XD Thanks everyone who replied!
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Yeah, like some airlines let you bring pet birds with you like you might bring a cat etc. Doesn't bother them the same way.
As for car sickness, I'll probably drive them around for a few hours in my car to see how they respond since it's been yeeeaaars, before I really consider sticking them in a truck for a month. I read that not letting them see out the side of the vehicle is key, which shouldn't be much problem in the back (and I tend to keep a beach towel wrapped around the sides and back of the cage anyway). I've also seen people say here and there that the more a bird flies, the less likely they are to get motion sick in a car.
My finches are in a flight cage and fly and fly and fly. When they invariably escape during any kind of cage transfer etc, it takes forever for them to wear out of flapping around in my bathroom (only room I do these things in, with the toilet seat down of course). They're strong little fuzzballs! Usually caged finches wear out fast, but my pair are a couple of workout fiends and they're in a big enough cage to do loops while also having four or five different perches (of different sizes and shapes, for toe health) that they land on whenever.