Stop.   Before you assume, that your smart movers are going to take your plants, you may want to make a phone call to confirm this, whether they are an agent for a van line, or an independent local moving company, call them first.  Most local movers will not move plants due to the fact that there is no way you can strap a plant in a truck, they do not want to get their equipment dirty, and liability on the life of your plant, and other reasons (insects, pests, mold, mildew).  If your mover does transport plants, this is usually pretty rare and they are specialized in doing so.  So what do you do, most movers will tell you to move them yourself.  So here are some tips for transporting your own plants, or if your mover does take plants then here are some tips for packing your plants properly.

1. Ceramic and glass pots are too heavy, so you may want to consider transferring your plants into a plastic, non breakable pot, usually like a month before you are going to make a move.  Your plant can be very sensitive to moving so you want to make sure it is nourished and adjusted to the container well enough, to make the trip.
2. Check for pests, mildew, mold, insects, anything that can infest your new home.  You can call the Dept of Agriculture to do an inspection, to ensure they are pest free.
3. Move only plants that have been in sterile soil.  You do not want to transfer any plants that have been outdoors due to the fact you may be bringing some unknown creatures to your new area.
4. If you cannot take your plants, you can cut the trimmings and replant at your new destination.  Cut the trimmings; place them in wet peat soil, or wet soil, and newspaper.  You can place these in an unsealed plastic baggy, so they have air to breathe.
5. Remember if you are using movers, you will not be able to insure your plants since they need to have a certain temperature, air to breathe, and enough water.
6. When packing your plants, you may want to use a wardrobe box for large plants and small trees, or a box like a dish pack, that is open on the top and cushioned on the bottom with plastic if a little damp, and paper in-between, so the plants are not tipping around and are in the box snug.  In the cold weather, when the weather is low in temperatures, you can put your plant in a plant sleeve, plants slide through them to provide the plant with warmth.  Moving smaller plants in a car or SUV can be a good option.
7. You should trim back any leaves or stems to condense space taken by the plant.
8. Make sure you are watering your plant on its regular schedule and giving it the appropriate nutrients, so it is strong for the move.  Remember do not overwater, or underwater as this can have different effects on the plant depending on the time of year it is and season.
9. You may want to consider pruning your plants to allow the best delivery for them and the stronger they will be, but do this about 2-3 weeks before you are going to move your plant so they are strong.
10. Remember when packing, you should pack you plants last, since they do need the most care, and will be confined for the shortest amount of time.  If you transport your plants in your car on a multiple day trip, make sure they get some sun every day.

When transporting your plants in the winter, remember indoor plants need warm temperature to survive, so move your plants quickly, avoid them touching the inside of cars, doors, etc since this can damage the plant.  Try to avoid putting your plant in a trunk, so they do not suffocate.

When you arrive at your destination, try to take your plants out of the boxes right away and avoid moving them too much, as too much movement is not good for your plant and they need to get acclimated to their surroundings.

Remember your plant will experience some sort of damage after being moved, since being in a new environment, they will need to adjust to different lighting, air, atmosphere, etc.