If your front action screw is tight and its stress free: it shouldn't be possible for it to move.
If tightening and loosening the the rear screw causes the barrel to shift, that means that the action is not in supported along its entire length, it is binding somewhere under load and thus not stress free and should be properly bedded.
If it were in proper contact the only way the barrel would be able to move is if the front screw was loose, otherwise there is no space in which movement could occur.
An example off of the internet where people measure the deflection that occurs from movement in the action with the change in torque on the rear screw.
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