As a general rule, most moving companies will know what to do with furniture that isn't safe to move fully assembled. Moving companies usually charge the dismantling and assembly service their usual hourly rate. Long-distance moves can sometimes change to a weight-based rate. When you're moving to a new home, it's important to coordinate the details with the moving company so that they know what is expected of them on the day of the move.
Most professional moving companies will assemble your furniture after unloading it from the truck and placing it inside your space. If your move is considered a long-distance move (more than 100 miles or from one state to another), moving companies will charge for the service based on weight. Using a moving company for furniture assembly has both advantages and disadvantages, depending on the type of move, whether local or long-distance. If you know that you have a lot of furniture that needs to be taken apart, you can always hire a maintenance staff or furniture dismantling service to do it.
You can significantly reduce the chance of your furniture being damaged during the move if you take it apart before the move. Some items are exempt from this rule, however. Most moving companies will refrain from dismantling antiques, pool tables, hot tubs, pianos, old clocks, and furniture that is attached with glue or nails. They won't risk it because they lack the time and carpentry knowledge needed to reassemble such intricate pieces.
Yes, many moving companies offer packing supplies in addition to packing services. Instead of going to the local home improvement store to buy supplies such as boxes, bubble wrap, wrapping paper and duct tape, you can ask the moving company if they offer these items for purchase. In order for your moving company to bill you properly and assign the right staff on the day of the move, inform them in advance that you need furniture dismantling and reassembly services. You're generally more concerned about strangers moving your furniture and personal belongings when you first move in, whether from your apartment, home, or workplace.
But when you have to move during the busiest time of the year, things can get considerably complicated. If you're moving locally, moving companies will be able to disassemble and assemble your furniture, but the move will take longer and cost you a little more.