Well... my experience is this: I like to use a carrier or a dog crate etc. that I can cover. Even just a sheet will work. It's less upsetting to the birds if they can't see odd things racing past them, someone walking by can't startle them, dogs can't scare them etc. Also, the sheet or whatever cover you use protects them from being in a draft. Being moved in and out of the crate is what creates the most stress, so I like to use one that they can fit in comfortably along with food and water, and then just leave them in there for the duration. In some cases, you will have to take the water out at times, because driving on the road can make it spill. I use the water nipples on buckets now, and those are fine - they don't leak - so if you use those, you can leave them in. Then all you have to do is check the food from time to time to make sure they are OK.
I tend to put perches in mine, but I make sure they are wired down very well to the bars of the crate so they are nice and sturdy. I like to use large round perches that they can not get their whole foot around, because I think it helps them to grip it better. Although I'm not a chicken, and maybe I'm wrong about that... Disclaimer: I have moved lots of chickens in my day but the birds I move the most are canaries. What I have done is simply do what I do for my canaries, on a larger scale for the chickens. So far it has worked out fine for me. Good luck with your new place!