When you're OOB, do your chakras come with you or do they stay with your body?
A good question but you are confusing the mental space with physical space. In physical space everything stays where it is. Its your consciousness that is moving, only your perception has changed, no longer restricted to physical world or sense perceptions. The word "out of body" is little misleading because its more accurate to say "free from body". I prefer the word AP.
Chakras are non-physical things, its very difficult to understand them in terms of physical location, because there is none. If you are a materialist rational man, you may logically conclude that they don't exist. If you are a new-ager kind and you have a little bit of experience with chakras, you may conclude that they are real , as real as your body. A spiritually evolved person will simply laugh....

When you are wide awake in physical, the sensations from the chakras seem to be coming from 7 locations in your physical body. If you cut there or do an xray or anything like that, you won't find anything unusual there. Your consciousness is totally focused in physical, you can't see beyond the body and conclude that they are in those 7 locations. There is a high possibility that you are wrong. At most we can say that something produced those sensations.
When your consciousness is focused in ethereal (pre-AP), these chakras seem more real and they are on your ethereal (RTZ) body. You start perceiving them there. Once you are astral or beyond, the chakras take on various forms depending on how you perceive them (and yourself). Sometimes nothing will be seen even if you are dead sure they exist. Sometimes people who know nothing about them, never heard the name, will be able to see them vividly and they appear as real as you and me. This makes you wonder, because the first person (s) who reported them must be having no expectation or intention to make them appear.
You can easily trace the chakra (or any other sensation) back to physical. They will shift places with your shifting consciousness. In the process you will be back to physical, wide awake , and e.g if its your heart chakra that you are tracing, it will be felt as a fast heart beat. If you take an ECG at that time, it will register nothing abnormal at all.