When you are moving, there are multiple times where your pet may get lost. It is even possible that they will be overwhelmed by the entire process and try to run away (and then instantly regret their decision). You want to be prepared for these situations so be sure that the tag on their collar is updated with at least their name and your phone number. If your pet has a microchip, ensure the information on this is updated as well.
Anytime you are moving with your pet you need to ensure that they have a clean bill of health. Moving can make any sicknesses worse so it is important to visit the vet. You should also ask for a certificate in case anyone along the way needs proof that your furry friend has had all of their vaccines.
If you are moving internationally, you will have to do careful research into the laws related to bringing a pet with you. At the very least you will need a health certificate but many countries will have specific requirements or additional forms that need to be filled out and some may even have a quarantine period for your pet.