What happens when I have a creature grappled and it dies?

Question

If I have a creature grappled and they end up being killed, do they fall prone? Does my character still gain half cover? And are they still subject to the grappled condition and could I still move around (at half speed) with them being dragged along?

Answer 1

You are no longer grappling a creature, you are holding an object¹

Grappling rules are out; encumbrance rules are in.

To your specific questions: prone is a condition of creatures² not objects.
If you hold the body between you and the enemy it should give cover, you can move them if you can carry or drag them subject to whichever option of the encumbrance you use.

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Notes: ¹ The Improvised Weapons rule points to a corpse being an object:
Improvised Weapons

Sometimes characters don’t have their weapons and have to attack with whatever is close at hand. An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin. (Basic Rules, p. 49)

² Conditions apply to creatures (PHB, Appendix A)

Conditions alter a creature’s capabilities in a variety of ways...

Prone

• A prone creature’s only movement option is to crawl, unless it stands up and thereby ends the condition.
• The creature has disadvantage on attack rolls.
• An attack roll against the creature has advantage if the attacker is within 5 feet of the creature. Otherwise, the attack roll has disadvantage.

Note: there is no reference to objects therein.

Answer 2

It depends on whether corpses are just objects, or are both objects and creatures

Whether a corpse counts as a creature is the underlying question which needs to be answered.

I have now realized this already has a question debating this situation here: "Is there anything that is simultaneously a creature and an object?"
The answer to that question is effectively what I argue in this answer.

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What we do know is that there has also already been a question here about whether a dead creature counts as an object, and it seems relatively agreed upon that the answer to that is "yes".
Thankfully, KorvinStarmast has pointed out in a comment that there is this section from "Improvised Weapons" (emphasis mine):

An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.

It states that improvised weapons must be objects, and then says that a dead goblin can be an improvised weapon. Thus dead creatures are at least definitely considered objects.

We also know that it is the designer's intent to have corpses be considered objects and not creatures from the now-unofficial tweets of Jeremy Crawford, such as this one:

A non-undead corpse isn't considered a creature. It's effectively an object.

And this one:

A corpse is an object. An unconscious creature is a creature.

But outside of the designer's intent, whether a dead creature (a corpse) also counts as a creature is more complicated...

I will first explain why the answer to that matters.

Why it matters whether a corpse is a creature or not

Under "Grappling", it states (emphasis mine):

When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple.

Thus we can conclude that at least initiating a grapple only works against a creature but not an object.

However, I believe that the requirements for initiating a grapple are required throughout a grapple. This is because if they were not there would be several oddities but the most apparent one would be that you could maintain a grapple without having a free hand.

It is never stated that you need a free hand to maintain a grapple, only that one is needed to initiate the grapple. I agree with this answer explaining that you do need a free hand to maintain a grapple.

This means that, if a corpse were a creature, you could continue grappling it; otherwise, you would be trying to carry/hold it, and so the rules on "Lifting/Carrying" or the optional rules on "Encumbrance" would apply.

If a corpse were still considered to be a creature, then you would continue grappling it and all the effects that come with that would apply. These include at least the following one (from the "Grappling" section):

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.

As the answer linked above explains, you need a free hand to maintain a grapple, and so you could not do things such as wield (attack with) two-handed weapons.

If corpses were not considered creatures, you would arguably (up to the DM) not need a free hand to hold/carry it.

There are a few other strange differences that would result from the grappled condition, which states:

• A grappled creature's speed becomes 0, and it can't benefit from any bonus to its speed.
• The condition ends if the grappler is incapacitated.
• The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.

All of these would still apply to the grappler and the corpse. The corpse's speed would be 0 and it could not benefit from bonuses to its speed (this probably does nothing).

You would let go of the corpse if you became incapacitated, which would not be the case if a corpse were only an object as the only thing that makes you drop objects is falling unconscious, which states:

The creature drops whatever it's holding and falls prone.

Also, you could not bring the corpse with you when hurled away such as by thunderwave which might be possible otherwise (you don't drop your weapons when hurled by thunderwave).

So in the end we have the following overall problem: If a corpse were a creature, then the rules on grappling could still apply.

What we know about whether a corpse is still a creature

There are spells such as revivify whose description states:

You touch a creature that has died within the last minute...

Or spells such as resurrection whose description states:

You touch a dead creature...

The problem is that if a corpse were an object, you could say that these spells don't work because they target creatures, and thus corpses must be creatures because clearly these spells work.

I believe that the wording of resurrection and similar spells shows that corpses are still creatures, but an argument can be made that they do not:

You could say that "a dead creature" and "a creature that has died within the last minute" are actually the descriptions of the objects - that "a dead creature" does not have to be a creature. (Similar things exist, such as that an "empty bottle of ketchup" is not actually a bottle of ketchup; it's just a bottle that is empty, which usually holds ketchup.) This would allow corpses to be only objects and not simultaneously be creatures.

I believe this to be a rather weak argument that the spells aren't targeting creatures but objects with "creature" in their name/description. There are spells (such as clone) which target "Medium creatures" but I would not call Medium creatures "objects" just because there is an adjective there.

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If corpses (dead creatures) are both creatures and objects then there is no real way of determining which set of rules to use: grappling, or carrying. It would be up to the DM in this situation.

I believe that something could be both a creature and an object because nothing in the rules, as far as I know, says this is impossible. There are also already some things which are neither a creature nor an object, such as the Circle of the Shepherd Druid's Spirit Totem, whose description states:

It counts as neither a creature nor an object...

We do not know whether a corpse counts as a creature, but we do know that it counts as an object.