Can Bigby’s Hand move creatures that are Grappled into the air?

Bigby’s hand can grapple via Grasping Hand. This allows:

The hand attempts to grapple a Huge or smaller creature within 5 feet of it. You use the hand’s Strength score to resolve the grapple. If the target is Medium or smaller, you have advantage on the check. While the hand is grappling the target, you can use a bonus action to have the hand crush it. When you do so, the target takes bludgeoning damage equal to 2d6 + your spellcasting ability modifier.

With a strength of 26, can it move creatures? Not only do you get the action, but you can:

When you cast the spell and as a bonus action on your subsequent turns, you can move the hand up to 60 feet and then cause one of the following effects with it.

So, after a grapple, could I move it 60′ in the air? Then next turn move it another 60′ and cause the crushing damage? At some point while it’s way up in the air, release the grappled creature for falling damage?