I would use the Tavern Brawler feat. This makes you proficient with improvised weapons and unarmed strikes, with unarmed strikes using a d4 for damage. Since you are attacking the target with your body and not a weapon, it would count as an unarmed strike and come under Tavern Brawler. Tavern Brawler also allows you to attempt to grapple the target as a bonus action after you hit them with an improvised weapon or unarmed strike, which then makes them easier for the rest of your party to hit.
Tavern Brawler gives you 1d4 total damage for unarmed strikes so you might want to try working something out with your DM so the d4s scale with the height dropped and weight of the dropee to better reflect the damage dealt to the target (e.g. 1d4 for every 10/20/50 ft dropped before reaching the target plus 1d4 for every 100 pounds of body weight or something similar). Some people have suggested 2d4 of damage in similar situations, one for the dropee and one for the target, however as you pointed out, a 2400lb barbarian dropping 400ft would probably do a lot more damage to the target than a single d4.
As for the damage to the dropee, I think it would be reasonable to have them take less damage than the creature they're landing on as otherwise there's not much point attacking in such a way - maybe roll the d4s for height and weight separately and have your character only take the height damage? These are very rough suggestions that would need to 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 flying speed. You can then drop them onto another creature as an improvised weapon.
Mechanics-wise, once you are grappling 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 flying carrying capacity and you can fly, you can fly and take them with you. You could then improvise a weapon out of the grappled creature that is now in the air using Tavern Brawler and use the shove action to launch them onto a different enemy on the ground. I would consider such an action to be an attack on both parties, since shove is an attack action and you are striking the creature on the ground with an improvised weapon.
The Grappler feat is not strictly necessary but comes in useful since you have advantage on all attacks made against a creature you are grappling - useful if the creature you're grappling decides to try and deal damage to you rather than escaping. An even better feat is Shield Master, which allows you to attempt to shove as a bonus action rather than a full action, which means that the whole manoeuvre can be executed in one turn! - grapple with your attack, use your movement to fly into the air, and then bonus action shove onto another creature below, all before the grappled creature has a chance to attack you. Tavern brawler allows you to attempt a grapple as a bonus action but then the grappled creature has a chance to attack you before you can shove them away.