With a large sized strength based Player Character (via Path of the Giant UA, Rune Knight or Enlarge/Reduce), it stands to reason that you could hit enemies with boulders, trees or other enemies if the following conditions are met:

• You grow to Large size and so are able to carry very heavy burdens, for example with 18 Strength you could carry around a Golf Buggy without slowing down

•  You have Tavern Brawler, and so are Proficient with Improvised Weapons, which can be virtually anything (Tree or Boulder) as long as your are happy with 2d4 damage.

•  You have Great Weapon Master and so -5 to hit with said Improvised Weapon and add +10 to damage rolls, for a nice 2d4+14 (if still using the aforementioned 18 strength)

Now if all of that is accepted, first what an exciting way to play, and second, a question: If you can use a dead goblin to attack a live goblin in the above fashion... could you grapple and then use a live goblin against another one also? I have found no rules to explore that question yet. With the closest interaction being a reflavouring of the following: "‘Cleave: If your player characters regularly fight hordes of lower-level monsters, consider using this optional rule to help speed up such fights. When a melee attack reduces an undamaged creature to 0 hit points, any excess damage from that attack might carry over to another creature nearby." (Dmg 272)

The rules for Grappling: https://www.dndbeyond.com/sources/basic-rules/combat#Grappling

Moving a Grappled Creature. When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.

You may only move a grappled creature when you move, and only by dragging or carrying, not by throwing.

I’m not sure what part you are responding to here? As I didn’t talk about movement with or throwing a grappled enemy? …Also in my example, the enemy was two sizes smaller.

could you grapple and then use a live goblin against another one also?

You can grapple but you cannot use it as a weapon. Because it is two sizes or more smaller your speed is not halved.

Quote from Houligan >>

You may only move a grappled creature when you move, and only by dragging or carrying, not by throwing.

Okay Houligan, I think I get where you are coming from here overall. Because the rules for moving with a grappled creature give two examples of how you can do that, you have interpreted that the only way to ‘move’ a grappled creatures is by carrying or dragging… though I’d argue that if you move with a creature (you can’t lift) in front of you, you’d be pushing them. Semantics sure, but perhaps helpful to highlight a non-exhaustive list. Your use of Only does not exist within the rules, that I can find.

So, acknowledging again that I know that Improvised Weapons refers to objects in it’s wording (which a dead body counts as). How would you rule a PC asking to hit an enemy with a different enemy?

1. A narrative explanation of two separate successful Unnarmed Strikes?

2. Two separate attacks, one an unarmed strike against the grappled creature, the second an improvised weapon attack against the other.

3. Two or possibly one Shove attacks? (Two enemies can end their turns in the same space just not willingly)

4. The ‘Improvise’ action and homebrew something

5. Say no

For grabbing a goblin corpse and attacking another creature you would not need to make a check to pick it up as a weapon. You would then make your attack roll as normal per Improvised Weapon rules.

For grabbing a living goblin to wield, I would definitely do Athletics to initiate the grapple onto it.
Then I think either a Shove or Attack as appropriate. The Improvised Weapon rules assume that your weapon is a static object. An unruly goblin is certainly not that. It would be well within your rights as a DM to oppose disadvantage on the roll because of how wiggly it is.