You may need to place a refundable deposit to reserve your move and the balance paid in full before delivery. The best international moving companies accept all forms of payment.
You will need to go through the claims process, and you can start by sending the international mover photos of the damage. The process can take up to a month.
Most movers aim for timely delivery, but there may be an "act of God" or something beyond anyone's control that can cause a delay. It's not common, but it's a possibility.