The correct statement is:
The object's weight changes, but its mass stays the same.
Mass is a measure of the amount of matter in an object and does not change regardless of location. Weight, however, is the force exerted on an object due to gravity, which varies depending on the gravitational pull of the celestial body the object is on. Therefore, when the object is transported from the Earth to the Moon, its weight decreases because the Moon's gravity is weaker than that of the Earth, but its mass remains constant.