How does one go about modifying a restraining order?

You need to immediately consult with an experienced family law attorney in your area. In Arizona, if she moved out of state, you should immediately file a petition for parenting time and legal decision making. Do not wait, as there is a time element for a court to exercise jurisdiction.

First you would need to request a hearing on the order of protection in order to attempt to have it set aside or at least to have your child removed from it. Then to obtain custody of your child, you would need to petition for divorce and file a petition for temporary orders, arguing that Arizona is your child's home state and that she should be returned.

You should probably hire an attorney to start a custody action. Of course, if the restraining order is justified, that will be difficult.

A key element in your situation is: how long ago did the mom and child move out? You likely need to file an action under something known as the "UCCJEA" (which stands for: "Uniform Child Custody and Jurisdiction Enforcement Act"), in the Michigan county where you all used to live as a family. Time is definitely of the essence in your case, so please don't delay.

You need a lawyer, badly. You may not want to pay a lawyer to help you with this, but I can assure you that without a lawyer you will be spinning your wheels and encountering brick walls almost everywhere you turn. Not many lawyers will admit this, but courts deal harshly and often unfairly with parents who have restraining orders or protective orders issued against them. You can fight this restraining order, but if you do so without an attorney's assistance your odds of success are slim. Talk to a lawyer. Find a good lawyer to help you, and resign yourself to the fact that you're going to have to spend some money to fix this problem.

In South Carolina, your wife can most likely do whatever she likes (in accord with the terms of the restraining order). However, if she moved out of state with your child without an Order giving her the right to do so, then although she has not violated a Court Order, it is likely that you could at least force her to come back to South Carolina through the Family Court and a custody / visitation action against her to enforce your rights to spend time with your child. I do hereby clearly advise against any reliance on this information as advice or the application of it to a specific situation without a more thorough consultation with counsel.