Ok but we're talking about escape velocity from the moon, for an object flung towards Earth. At a certain distance from the moon, along such a trajectory, Earth's gravity has more influence on an object than lunar gravity, so it starts being pulled away from the moon without needing to expend more energy than is needed to reach that point.

My question is whether the escape velocity from the moon alone greater than the velocity required to reach the point where Earth's gravity can just take over.