Moving bases can be a little bit of a pain, but it's completely doable. We recommend moving bases a little as possible because depending on how much you have built, it will take you a lot of time to dismantle everything and move. You can also only have three active bases after reaching level 10 and 15 respectively. Here's the general process if you do choose to move bases during the game.
The first thing you'll need to do is go to each structure by your Palbox such as ranches, breeding farms, Pal beds, and egg incubators and dismantle them. There isn't a way to pick them up individually, but you'll instead get all of the initial materials you used to build them. When you dismantle your Palbox, any working Pals will get sent back to the box.
After you have dismantled all of your base structures, you're pretty much done and ready to move on. Your non-work related structures will stay in tact in case you want to leave them there. Make sure to grab anything from your chests that you want to keep. Now you're ready to venture forth and find a new spot to establish your base and build everything again.