As a college student, I have to move Godric between all the long breaks of the school season. At first I was worried that the movement would stress him out and mess up the tank ecosystem, but he really doesn't seem to care and I use the same tank ornaments and everything so my helpful bacteria friends don't all go to hell each time (haven't had any problems with ammonia or nitrite).
To transport Godric, I use a long Tupperware tub, about 6 inches deep and 12 inches long (not exact measurements; just estimating) and I keep it topped with a solid plastic lid that has air holes cut into it. I also use the same container to do some of his feedings, so he's used to being in it. Maybe it doesn't affect much, but I feel like it minimizes the stress because he's familiar with it.
To avoid the Georgia heat, I usually put his container in a cooler that has a thin layer of ice cubes on the bottom, but if your ambient temperature is low enough then I wouldn't worry about that.
Just try to minimize the time your axie is out of the tank as much as possible, even if it's seconds. Godric's tank is the last thing I pack and the first thing I set up again, always. Keep the axie container as stable as possible; don't let it jostle around if you can help it.
Otherwise, moving your axies shouldn't be too much of a problem. Just plan your steps out beforehand and all should be fine
Also, a quick edit: I personally would not put more than one axie to a container. The close proximity is just too much of a risk for one of them to be nipped. Travel is stressful enough without your tankmate taking a bite out of you >.>