If it can get hot and burned, a moving company can't stand it, either from a legal or safety standpoint. If you plan to move ammunition, you can do so in your own vehicles, as long as you comply with all state and federal laws while doing so. This applies whether you are moving the weapons yourself or hiring a moving company. By law, carriers are allowed to carry firearms, but not ammunition, in a moving truck.
However, moving companies do have the right to establish policies stating that they will not carry firearms. If you encounter this situation, you can find another moving company or move weapons yourself. For the actual packaging of firearms, you need a box slightly larger than the firearm, bubble wrap and peanuts to pack. Start by wrapping the firearm in the bubble wrap and then place it in a box full of peanuts to pack.
I recommend it for every firearm to avoid damage. Another option is to use a padded case for firearms. Pack your ammo separately as an additional precaution. As far as ammunition is concerned, it should be stored out of reach in the trunk or other locked container.
If you want to move from home and also move weapons and ammunition, pack all the required permits and registrations. However, moving companies are legally prohibited from transporting ammunition in their moving trucks, even if the ammunition is not loaded into the firearm. Carefully load it onto a mobile platform and transport it the same way you move a refrigerator, closet, or any other large household item. It is then a matter of following the procedure for the firearms by unloading them, packing them properly and making the crew on the move aware of the firearms at the time of the move.