There is no general rule concerning involuntary movement and whether it provokes AoO's, so let's look at some exceptions that might prove one.
Pull is an ability that involuntarily moves an opponent, and explicitly does not provoke:
Pull (Ex): A creature with this ability can choose to make a free combat maneuver check with a successful attack. If ...
Every creature within the area of the Black Tentacles spell that loses their grapple check is grappling. So the combat modifier for ranged attacks versus a grappling creature ("Roll randomly to see which grappling combatant you strike, defender loses Dex bonus to AC") applies.
Only if the tentacles were incorporeal, illusory, telekinetic or of ...
You are able to shove or grapple while prone.
The Basic Rules outline the prone condition:
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 ...
Consider using optional DMG actions
In the Dungeon Master's Guide there is a section for extra action options one could include in a game and one of them is Climb onto a bigger creature (DMG p. 271) (emphasis mine).
As an alternative, a suitably large opponent can be treated as terrain for the purpose of jumping onto its back or clinging to a limb.
This is ...
This is Improvising an Action, so the DM decides
But a Strength (Athletics) contest is most appropriate, although probably only as an action on subsequent turns
The first half of the plan, where the dragon grapples, lifts and drops characters from a height, is a valid tactic, covered by NautArch's answer and this question. Using solely the rules as written, ...
The dragon action of picking up and dropping is a very cool one, but it isn't necessarily the best move always for the dragon. If you'd like to play the dragon like this so it's 'easier', it's all good!
So, let's start with the mechanics. For general purpose information, you can see the grappling rules which cover most of what I say below.
Assuming the player is holding on, there are at least two possible actions that the dragon can take on the following turn to break free from the player's grasp:
Shove action (a reasonable interpretation of the shove action in this case is that the player would not be holding on anymore after a successful shove and would fall from the height)
Using the ...
Even if your DM rules that the rope can hit from 20 feet, your reach is still 5 feet and grappling uses a free hand.
Using at least one free hand, you try to seize the target by making a grapple check instead of an attack roll: a Strength (Athletics) check contested by the target's Strength (Athletics) or Dexterity (Acrobatics) check (the ...