The short answer is yes. In Georgia state statutes provide for service and jurisdiction over divorces even when the defendant lives in another state. This is possible as long as the party filing has lived in the state of Georgia for six months prior to filing the action. So if your spouse relocates and tells you that you are not getting a divorce unless you come to their state they are incorrect; however, the longer answer is a bit more complicated.
If you have children and need to deal with issues of custody, visitation or support you will not be able to resolve all of these issues. Georgia can exercise jurisdiction over your divorce but cannot exercise jurisdiction over the children if these children have become residents of the foreign state. Once they become residents of the foreign state you will have to submit and file in that state. Time is of the essence in these situations. There are applicable statutes, but the most relevant is the UCCJEA, which defines the home state as that state in which the child lived six months prior to filing the action. So you see, if your spouse leaves the state with your children and you delay action you may find yourself bearing the expense of travelling to the new state and, more significantly, convincing he court in that state that the children need to come back to Georgia with you after establishing residency in the new state.
The advice here is when you are dealing with a roving spouse do not delay, make certain you know where they are going and file the papers.