I was inspired by this question, where the answers propose improvised weapon and unarmed strike rules to ajudicate the effect of a monster picking up and tossing a (in this case unconscious) creature as a weapon. This got me thinking about whether some published monster has an attack with specific rules for how to pick up and toss another creature, including weight/size limitations, effects/saving throws for the tossed and targetted creature(s).
Stone Giant (variant)
In the adventure Storm King's Thunder we are presented with variants to the giants in Appendix C.
The Stone Giants receive this new feature:
Fling. The giant tries to throw a Small or Medium creature within 10 feet of it. The target must succeed on a DC 17 Dexterity saving throw or be hurled up to 60 feet horizontally in a direction of the giant’s choice and land prone, taking 1d6 bludgeoning damage for every 10 feet it was thrown.
Note that the Cloud Giants also have this feature, with higher save difficulty and damage.
Krakens from the basic rules have a similar feature, but it requires the creature to be grabbed first.
Fling. One Large or smaller object held or creature grappled by the kraken is thrown up to 60 feet in a random direction and knocked prone. If a thrown target strikes a solid surface, the target takes 3 (1d6) bludgeoning damage for every 10 feet it was thrown. If the target is thrown at another creature, that creature must succeed on a DC 18 Dexterity saving throw or take the same damage and be knocked prone.
This feature is also given to other Kraken variants, like Malformed Kraken, Young Kraken and Slarkrethel.
There are at least two monsters in the Monster Manual which have this kind of ability: The Kraken and the Air Elemental.
The Kraken can throw a grappled creature of size category large or smaller. The creature takes damage depending on the distance if it hits something solid.
The Air Elemental can fling creatures in its own space using its Whirlwind, if those creatures fail the saving throw. Whirlwind always deals damage but the target is only thrown on a failed saving throw in which case damage depending on the distance is also dealt when something solid is hit.
If you want to homebrew creatures that can toss characters or give this options to existing creatures, you need to first check what the consequences are. If characters can be thrown into a chasm or something similar, this can be a problem but if it is simply another form of attack dealing damage, e.g. if a character is thrown against the dungeon wall you can simply balance this through the damage. That is consider the expected damage dealt by throwing when determining the challenge rating. If throwing replaces an attack which would otherwise deal the same or more damage, you may not even have to consider it for the CR (barring corner cases like the chasm mentioned above).