As a general rule, movers do not typically offer temperature-controlled moves. It is an uncommon service because it is not commonly requested. However, if you have multiple belongings that all need to be moved with specific temperatures and humidity levels in mind, here is how you can manage this.
Ask the Moving Companies If They Are Willing to Help
Most moving companies are generally willing to assist customers with special requests. Ask each moving company you contact if they are willing to make special accommodations for moving your temperature-sensitive items. If they are, then the next step is to determine how you will approach this problem.
Keep Things Cool and Dry
More often than not, items need to be kept cool and dry rather than in hot, warm, and/or humid conditions. If cool and dry is what you need, then the next step is to select materials that will accomplish this. Your best option is to place your objects in crates with packets of silica gel dessicant, which absorbs humidity and moisture rapidly and draw it away from your objects. To keep things cool, you should look into large bricks of dry ice, which can be placed in boxes and/or styrofoam crates and loaded onto the moving truck a few hours before you load your belongings.
Clear the Use of Your Chosen Environmental Control Materials
The dessicant packets should not be a problem at all for the movers since dessicants are commonly used in packaging and shipping. The dry ice may be a problem, which is why you should ask the moving company if they mind transporting several pallets of well-secured dry ice on their truck. If the ice is well-contained and is unable to slide around inside the truck, it is likely that the moving company will not have a problem with it. If you also agree to remove the ice from the truck at your destination, they should be willing to accommodate you.
Order and Place the Dry Ice on the Truck
If this plan is a go, then be sure to order all of your needed dry ice about a day or two before you move. Dry ice has a tendency to revert back to its gaseous state if it is left exposed to any temperature above its solid form (which is pretty much any temperature at all). If you have the ice delivered and placed out of the way of the sun the day before you move, that is a good idea. Place your dry ice on the moving truck right away on moving day and then get all of your belongings on the truck as quickly as you can so that the truck can start to cool off when the doors are shut.
Contact a moving company to secure movers for the day you need them, and be sure to ask for any special accommodations you may need.