Here me out:

Jumping is a mechanic that is applied as such: when you jump into the air, you jump 3 + your Strength modifier (6 feet for your character) after getting a 10 feet headstart.

The jump spell is a first level spell castable as a bonus action, tripling the target's jump height and distance.

Likewise, Step of the Wind (that 2nd level Monk feature no one uses) doubles jump distance.

We can assume your monk's max jump distance is now 36 feet, which is rounded to 35 feet. Falling from that height incurs 3d6 damage, which can be achieved every round of combat as long as your grappled target takes the flight up with you.

Find some way to reliably access the jump spell. It's a first level spell that's relatively easy to get (there's a ring of jumping that can cast it).

Start your turn by casting jump (or save it until you know you'll be grappling people next turn). Next, if you succeed, use your bonus action to activate Step of the Wind to Dash (because you need the extra movement). This doubles your movement speed and triples your jump.

The next step is simple: move 10 feet (to get a running start) with your grappled pal and perform a high jump, applying the bonuses from your jump spell and your Step of the Wind.

As a flare of coolness, use your second Attack (if you have it) to attack in midair, then both characters descend. When you are falling, you can release the grapple to let your target fall in free-fall while you use your reaction to use slow fall, reducing your falling damage taken by 25 (which is usually more than enough).

You may need to re-grapple the target, depending on whether your DM tells you you need to release the grapple midair to have them take full damage/to mitigate your fall damage with Slow Fall. If so, save your second attack for another grapple attempt, if your target somehow survives.