If the vehicle is on a trailer then it does not need a current registration. If the wheels are not on the ground it does not have to be registered. Still make sure you have the old registration and title to prove it is yours. A few states (not many) require you to stop at the truck scales on entering the state if you have an unregistered vehicle. Utah is one of them. You follow the rules of the last state where the vehicle was titled or registered. Some states offer a 5 or 10 day drive away permit to take an UN-inspected / unregistered vehicle out of state. This is not necessary in your case because the vehicle is on a trailer. Liability Insurance is also not necessary since you are not driving the vehicle on the highway. The trailer will need a tag based on where it is registered. If it is a U-haul it could be registered anywhere.
If there are no signs up at the first truck scale when you enter a state stating "unregistered vehicles must stop", then just keep rolling. The last time I stopped at one, the guy at the scale asked where the unlicensed motorhome was from. I said "Michigan". He said "have a nice day, Michigan does not require a plate to transport a vehicle between states". He never asked for any paperwork.
2005 White 2wd Z71 WT