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Having a vehicle that has an auto loan on it generally doesn't prohibit you from shipping it.
You can take it anywhere in the contiguous United States in most cases. However, if you are shipping to Hawaii, Alaska, that may change. So, be sure to make a quick call to your lien holder / bank and ask if it's ok.
If going to Puerto Rico, you will definitely need to have a notarized letter . This letter needs to say we can ship it there. This is needed even if you are told verbally it is ok.
In many cases there shouldn't be a problem with you sending the vehicle, but it should not be assumed. This can be especially true with smaller financial institutions or local banks. So the lien holder / bank can definitely deny you from shipping.
Remember, even though you have the registration of your financed vehicle, you don't own it. The bank owns the vehicle and they have the final word.
You have to keep financed or leased vehicles insured. Even if shipping to another state, be sure that the insurance requirements are the same. Also make sure you can afford the insurance if moving to a metro area.
Once you are ready, have checked with the lender, the next step will be to get an auto shipping quote. From there we handle the rest for you. You'll get your vehicle situated at its new home quickly and safely.