Arrow to the wing is at least some rational explanation, but a grappling a winged opponent in the standard sense, while it has the desired effect, I would rule that you fall along in this case, and the moment you hit the ground you take the same falling damage. If you opt to stay in your position after the grapple then one of two things happen: You let go and the creature catches itself or you carry the creature which requires a strength check or you let go. If you carry the creature then it canot fall and take damage from it.
Ok let us do dragon grappling Coroc style:
Player has got the ability to fly somehow and is no wussie and decides to impair the dragons wings by a grapple.
1. Acrobatic check to get into position - it is no simple attack but a targeted attack so either that or disadvantage.
2. Opposed str check
3. You impair the dragon
4. the dragon falls x feet / per round
5. as part of his attack the dragon either toasts you on his back (he is immune vs his breath) or tries to get loose, each round in the air so again saving throw with disadvantage, because you grapple you cannot move out of the way of the breath or opposed str check
6. You manage to survive 1.-5. somehow the dragon hits the ground. you have to clinge on to the dragon until the impact else the dragon recovers. The dragon takes falling damage. The dragons body is hard as steel due to scales and reptile muscles. You take falling damage.
7. Unless you are a 800 HP bear totem barbarian we got 1 very angry dragon and a hero who starts making deathsaves, ah no forget about that, the dragon takes it personal and breathes again and does his full attack
Edit: I forgot, each ranged attack or spell by your buddies airborne or not, has the chance to hit you instead of the dragon, but you get advantage on that one because the drago nis much bigger and more likely to be hit