While many people leave their plants behind, many treasure their plants that they will bring them no matter where they go. If this is the case, you should start preparing your plants weeks before the move. Even plants feel the stress too. And so, you should start pruning the plants a couple of weeks before the move. If you are not sure how to do it properly, you can search online, read a book or ask an expert.
You should also get rid of all pests breeding in your plants about a week before the move. To do so, place the plants inside a black plastic bag and apply pest strip or bug powder. Simply close the bag overnight to kill the pests in your plant. Then prepare the plants for transport a day before the move. Start putting them in cardboard boxes.  But to keep them in place, place some damp newspaper at the bottom. To keep the leaves from drying cover each with wet paper towel. If you are only bringing cuttings of the plant, put them in a plastic with wet paper towels as well.
On the day of the move, you can now close the boxes. But punch some holes to allow air to enter. Then label them so the movers will not take the boxes. Bring the boxes with you and unpack them as soon as you arrive your new home. Do not place them in direct sunlight though. Let the plants gradually adjust to their present climate and temperature instead.