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.