Moving to a new home doesn't mean that you need to be parted from your pot plants, regardless of how long the journey is. Small plants, and even trees and shrubs can easily and safely be moved as long as you do a small amount of planning. Most removalists can easily accommodate live plants too, as long as they have been properly prepared.
It's unlikely to be a problem for your removalists to transport your plants, but you should discuss the matter with them first so they're prepared. They might wish to stock protective material, such as plastic drop sheets to protect your other possessions from any spilled soil or water that might seep from the pots.
Your larger shrubs and trees should be pruned prior to being packed. Removal of this excess foliage will make the plants easier to pack, and they will also expend less energy to maintain this foliage once it has been removed. Of course, you can't simply remove all branches, and so the remaining branches (if pliable enough) should be pushed back towards the trunk before wrapping a loop of garden twine around the remaining foliage. This should then be knotted into place, effectively reducing the mass of the plant.
This foliage bundling will generally not be necessary with smaller plants, and these can be packed into a cardboard box that has sufficient room to be closed without damaging any leaves and flowers. These smaller pots should be wrapped in newspaper prior to being placed in the box to prevent breakage while they're in the back of the removalists truck. Depending on the length of the journey, a slow release fertiliser might be a good idea, as well as watering prior to packing them.
Larger plants can also benefit from some slow release fertiliser, and if you believe any of them will need water on their journey, you can make your own gradual watering system from a soft drink bottle. Be sure to choose a bottle commensurate with the size of the pot and you might not need to use a lot of water. It can also be good idea to secure the upturned bottle with a garden stake and some gaffer tape. Larger plants should have their pots wrapped in bubble wrap or newspaper and then placed into a heavy duty rubbish bag to prevent soil spillage (and to contain any water that might seep from the base). This bag can then be loosely tied around the trunk of the plant.
With a little bit of preparation, your plants will emerge from the removalists truck as healthy as they were when they went in.