That would be even worse. Let a strong gravitational force go through the wormhole and drop something near it (not through it).
Then move the wormhole (and the gravitational force) elsewhere, lift the object, and bring the wormhole back.
Normally the thing that generates gravity has inertial mass, so it costs energy to move. Not with a wormhole. If you speculate that it should have inertial mass, then how much?