If someone has a valid restraining order in one state and then moves to a new state, they do not need to get an entirely new order. That person simply has to register the existing order with the court or police department in their new state and it will take effect immediately. For example, if someone obtains a restraining order in Florida, and then moves to California, they need only bring a certified copy of the order to a California court, along with a registration form, to obtain a new restraining order.
Again, a restraining order is valid everywhere in the United States, and the named party will be protected by it no matter where they are. This is due to a section of the Constitution called the Full Faith and Credit Clause. If someone is concerned that they may not receive protection from a restraining order, they may consider keeping a copy of the restraining order with them to show to police in the event anything happens.
If you are moving to a new state and you have a restraining order against someone else, it may be wise to consult a criminal lawyer, especially if you have children. Speaking with a lawyer may help you understand your options and recommend the best actions to take. A family lawyer can also instruct you as to the steps to take to register your existing restraining order in your new state.