Instead of trying to move the row of data from one table to the next, why not add an extra field (an Owner field) to the cars table that contains the current owner's User ID. Then when a user "absorbs" a car, you will UPDATE the car's row so it's Owner field contains the user's id. If the user "drops" a car, you will UPDATE the car's row so it's owner field is empty.
On the left side of the screen, you display all the car's that have an Owner field equal to the user. On the right side of the screen, you display all the car's that have an Owner field that's empty.
It's pretty simple actually. Hope that helps. If you want some additional code examples, come to my website AaronZoller.com.
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