First off, HUGE shout out to TidoMan because without his glitch for insuring the cop vehicles this method wouldn't be known either so props to him!
I made a video that shows you exactly how to move around the cars in your garage. You will need to permanently destroy a car in order to make your first move but that should be it so chose your sh*ttiest one. Cheers! This is also the same method used to insure vehicles such as cop cars and other police vehicles. When you pull the car back in the garage it will be insured regardless of what it is. I have even insured Buzzard choppers doing this!
If you have a car that is duplicatable for a friend (they get in car, start up job, drive into garage, exit job) then I suggest making a dupe for them and using that as the initial car to destroy. That way when you have all the cars in order you can grab that car back from your buddy (dupes it back to you) and use it to replace the last clone you end up with so you end up with all ten cars you originally had. This is what I did with the space docker.
I strongly suggest using an invite only session. Make sure you have no currently destroyed cars that can be replaced with Mors Mutual. If you do then you need to call and get the replacements and make sure they are all back in your garage.
1. Determine the spot where you want the desired car to be moved to (must already be a different car in this spot thats also insured).
2. Grab the car that is in that spot and drive it outside of the garage.
3. Blow up the car and lose the police if you get a wanted level (usually will).
4. Enter the garage again on foot and get in the car that you are trying to move (don't move it/hit gas AT ALL).
5. Call Mors Mutual on phone while sitting in drivers seat in the car you want to move.
6. Once the list appears for cars you can recover (don't actually select the car you blew up) drive the car out with menu still up.
7. Once the car starts moving and the screen turns black start spamming the A button (or X for PS3).
8. When the cutscene ends and you drive out of the garage you will see in the corner that the claim has been processed.
9. Drive car back into garage.
When you drive the car back into the garage it will park in the spot where the car you destroyed used to be (thus moving it). You will notice however, that an exact clone of this vehicle will still be parked in the spot it originally was in. This random copy is the car you will take out and destroy next thus setting up your next move. Rinse and repeat. You will not be able to ever recover the original vehicle that you had to destroy which is why I suggest starting with a car that you can successfully dupe for a friend (see above) that way when you are all done you can have him dupe it back to you which can be used replace your final clone (since you will always finish with two of the same car).