Summarize:

Once your car crosses a national border, it becomes a national law enforcement issue as well as a local police issue. This is especially important if you recover the car yourself, because you may be detained at the border.  Notify the local police department as soon as your car is recovered. They may have an officer come and meet you at the border to process the recovery of the car. At the border, let the border agents know that your car was reported stolen and has been recovered. Provide identification as well as proof of ownership to ensure you aren't detained any longer than necessary, or that your car is not impounded. If the police find your car, they'll take it to the impound lot for processing. To get your car out of impound, you'll have to provide proof of ownership and pay impound fees, which may be several hundred dollars.  The detective who found your car will give you a phone number for the impound lot to which your car was towed. Call them ahead of time and find out how much you owe in towing and impound fees, and what methods of payment are accepted. Ask if the car is in drivable condition. Otherwise, you may need to make arrangements to have a tow truck meet you at the impound lot so you can get your car to a mechanic. As soon as possible after you hear from the detective that your car has been recovered, call your adjustor. They will update your claim and let you know what you need to do if your car is damaged. Typically the insurance company will tell you to take your car to a mechanic to get it inspected, even if there doesn't appear to be anything wrong with the car. Take them up on this, otherwise you may end up paying out of pocket if a problem occurs later, even if it happened as a result of damage from the theft. Use a flashlight, and look all through the interior of your car, between and under seats, and in all storage compartments. Look for items that do not belong to you, which may be evidence of another crime. If you find anything that doesn't belong to you, notify police immediately. Don't touch it or move it, and don't move your car until police come and process the evidence. Even if there doesn't appear to be any damage done to your car, it's a good idea to take it to a mechanic for an inspection. There may be damaged parts that won't affect how the car runs until later.  Mechanics will inspect your car thoroughly and provide a written estimate of repairs to be completed. If you're paying for repairs yourself, you can choose which you want to do now and which you want to save for later. The mechanic will tell you which repairs must be done immediately. When repairs are coming out of your own pocket, you may want to get more than one estimate so you can make sure you're getting the best deal. If repairs are being covered by your insurance, go through the estimate and be honest about any problems that your car had before the wreck. For example, if the mechanic has included repairing scratches to the paint on a door, but you know the scratches have been there for months, let them know that doesn't fall under the insurance claim. It may seem like a minor thing, but letting insurance pay to repair something that wasn't part of your claim is insurance fraud.
Contact national authorities if your car is recovered in another country. Retrieve your car from the police impound lot. Let your insurance company know your car has been recovered. Search the car's interior carefully. Get an estimate for repairs.