I would use the Tavern Brawler feat. This makes you competent with improvised weapons and unarmed hits, with unarmed hits using a d4 for damage. Since you are attacking the target with your body and not with a weapon, it would count as a disarmed blow and would remain under Tavern Brawler. Tavern Brawler also allows you to try to attack the target as an extra action after hitting it with a makeshift weapon or an unarmed hit, which makes them easier to hit for the rest of your group.
Tavern Brawler gives you 1d4 of total damage for unarmed hits, so you should try to solve something with your DM, so that the scale d4s with the dropped height and weight of the dropee better reflects the damage inflicted on the target (for example, 1d4 for every 10). / 20/50 feet fell before reaching the target plus 1d4 per 100 pounds of body weight or something similar). Some people have suggested 2d4 damage in similar situations, one for the dropee and one for the target, however, as you noted, a 2400 lb barbarian throwing 400 feet would probably do much more damage to the target than a single d4.
As for the damage to the dropee, I think it would be reasonable for them to receive less damage than the creature they are landing on since, otherwise, there is not much point of attack that way, maybe they roll the d4s for the height and the weight separately and does your character only receive height damage? These are very approximate suggestions that should be refined further.
If you have a character that can fly, taking the Tavern Brawler feat will allow your character to pick up a creature and fly with them at half the speed of flight. Then you can drop them on another creature like an improvised weapon.
As for mechanics, once you're fighting a creature, you can use your action to move at half your movement speed, effectively dragging the creature with you. If the creature is less than your flight load capacity and you can fly, you can fly and take them with you. Then, you can improvise a weapon from the creature with hooks that is now in the air using Tavern Brawler and use the push action to throw them against a different enemy on the ground. I would consider such an action to be an attack on both sides, since shove is an attack action and you are hitting the creature on the ground with an improvised weapon.
The Grappler feat is not strictly necessary, but it is useful, since you have an advantage in all attacks against a creature that you face; useful if the creature you are facing decides to try to hurt you instead of escaping. An even better feat is Shield Master, which allows you to try an extra action instead of a full action, which means that the entire maneuver can be executed in one turn! – deal with your attack, use your movement to fly in the air, and then throw an additional action on another creature that is underneath, all before the trapped creature has the opportunity to attack you. The tavern wrestler allows you to try a fight as an additional action, but then the creature that has a fight has the opportunity to attack you before you can push them.