It is a slightly complicated procedure and basically the hamster has to take priority! I've only done it once. We moved out of one place and into the other place in the same day.
For move out. Hamster in pet carrier in the quietest room of the house with a blanket over the pet carrier - away from all the noise and bustle of removal men, packing up last things etc. Removal men also took hamster cage and had strict instructions to put it on the van last, and bring it out first - and to handle it carefully and not break it (they still broke the base!).
Once everything packed up and gone, we headed off to the new place with hamster in pet carrier.
When removal men got there, left hamster in pet carrier in quietest place in the house again, with blanket over. The hamster basically spent a day in the pet carrier and didn't move into the cage until evening. He was fine - he just slept.
Hamster cage set up and hamster moved into it that first night. (see edit in post below). Not the final location for the cage but left it there for two weeks to avoid further upheaval (while popping in and interacting every evening).
Then took weeks to unpack everything lol!