People travel with projectors all the time. Heck, they are shipped on the slow boat over from China, then UPS'd right to your door. Not likely they are using pillows along those joy rides.
As long as you take some care with their transport, they will do fine. Heat/cold will not affect a projector very much when it is not actually on. Just give it a couple of hours to acclimate to the place where you finally end up using it.
For example, if you leave it in your car and your car is 100+ degrees out in the sun, don't bring it into the A/C home and turn it on right away. That could hurt things. If you give it a few hours, you won't have any issues. Same is true in reverse. Fine to leave it in a car when it is 10 degrees outside, but inside your 70 degree home, give it a few hours to warm up before you turn on the projector.