Can a monster with a flying speed grapple a PC, use half its movement to fly directly up then drop the player in the same turn?
Would any 5th edition core book ruling prevent me from doing any part of this?
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The rules for moving a grappled creature (grapplee) state (PHB 195, emphasis mine):
When you move, you can drag or carry the grapplee with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
Moving over to where "drag" and "carry" are defined (PHB 176). Dragging loads in excess of your carrying capacity reduces your speed to 5 feet, so you aren't going to get much height there. Carrying only works on loads up to your carrying capacity (15 * Strength score * size class modifier). Note that the flyer's load includes the grapplee's load, their weight (and they laughed when I insisted my Dragonborn weighed 300 lbs), and anything else the flyer might be carrying.
Once you're aloft, dropping is trivial (PHB 195):
[...] you can release the [grapplee] whenever you like (no action required).
By the plain reading of the rules a flying monster can grapple in this manner. Provided the conditions are met. Halved speed, size requirement, etc.
From page 74 of the 5e Basic Rules for Players
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.
The dropping of the character is handled by the rules of interacting an option on page 70 of the 5e Basic Rules for players. If you can pick up a dropped axe, releasing a character from a grappled to let him fall is certainly covered under a interaction.
The monster will not get an attack of opportunity. The fall of the character is a not a result of the character's action, movement or reaction.
From page 74 of the 5e Basic Players rules.
You can avoid provoking an opportunity attack by taking the Disengage action. You also don’t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction. For example, you don’t provoke an opportunity attack if an explosion hurls you out of a foe’s reach or if gravity causes you to fall past an enemy.
From the PHB, page 195:
Moving a Grappled Creature. When you move, you can drag or carry a grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
That's all it says, there's nothing about flying (or any other type of special movement) in there. You can carry a creature with you if you're flying, if you're climbing, swimming, or even burrowing.
As for dropping it (from the same page):
[...] and you can release the target whenever you like (no action required).
So yes, you can grapple your target with an attack, fly up with your movement, then drop them for free all in the same turn.