Grappling as a combat maneuver has specific consequences, but the case of jumping onto a pegasus/hippogriff is more akin to "mounting".

It is the DMs prerogative to resolve it with skill checks, rather than a special "attack" action.

This would fall under the trope of taming a wild stallion, with higher stakes.

Quote from Wysperra >>

Quote from DxJxC >>

Quote from Wysperra >>

Does this also work to prevent something from taking off? 

It works against a creature that has multiple movement types?

 Have you seriously never read grappling rules in the 2.5 years you've been on the site? It is in the basic rules under making an attack - melee attack.

 I have read them but there are still situations that I need to wrap my head around. 

So a PC A is flying in the center of a burning tower. PC B leaps to grab PC A so they can both fly out. But if they do that, they fall to the ground and die.

PC grabs hold of pegasus or griffon, and poof, auto capture. No dramatic, flying creature takes off with you hanging by the legs.

PC riding flying dragon leaps upon flying hippogriff, splat. 

 Not exactly. If they are intending to grapple and hinder movement, then that is how the rules go.

If they are wanting to get carried or mount, that is an entirely different set of rules.

It all falls on the intent of the non-flier. Grabbing and grappling are not necessarily the same thing.

Thanks!

I appreciate the prompt and helpful replies =)

I think I will consider that instead of breaking a grapple, and being free,  the subject has the option for dragging the attacker along. Like when a cowboy lassos a stallion and the stallion runs away but the cowboy hangs on.

Quote from Wysperra >>

Thanks!

I appreciate the prompt and helpful replies =)

I think I will consider that instead of breaking a grapple, and being free,  the subject has the option for dragging the attacker along. Like when a cowboy lassos a stallion and the stallion runs away but the cowboy hangs on.

Grabbing on to something is not the same as grappling. There could be any number of ways and methods to attach one creature to another, but not all of them result in the Grappled condition. The special Grapple attack and any spell or ability which applies the condition is very specifically trying and succeeding at keeping that target from moving. It generally involves a contested strength test or a strength save, because any target strong enough can just brush off such attempts.

That said, a house rule to allow a grappled creature some ability to move without breaking the Grapple - as an action with an opposed strength check - might be feasible. But it will probably mean players will use grappling options even less than they already do. 

So is it possible to grab onto the huge dragon but not grapple it? I guess that's a GM thing.

Quote from Wysperra >>

So is it possible to grab onto the huge dragon but not grapple it? I guess that's a GM thing.

 There are definitely rules for that, but they are optional.

So, if I understand the rules, according to RAW, if one were to successfully attack a flying creature that does not have hover, with a bola, then it is grappled and starts falling from the sky? I mean, a bola could wrap around a foe's arm or their feet, and so while the bola has been accurately thrown, how does that automatically mean the target is grappled—especially a flying one?

I don't think there are official 5e rules for bolas.