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I can't find an answer to this in any of the books. I am playing a Halfling Monk. I have a half-orc barbarian on my team. He picks me up and throws me at an enemy. He gets a 21 because of strength bonus. I get a punch lined up and get 16.
Do I get a bonus on damage because of air speed? How does this work?
Your half orc used you as an improvised weapon, so when you hit you do 1d4 damage (rolled by the half orc player).
A comment asked for clarification: Is that damage to you or the target? Or Both? (@NautArch) To the target that the half orc threw the halfling at.
Your Halfling is less massive than a dead goblin (cf. below), so it approximates the effects of a dead goblin sufficiently for the purposes of this attempt.
While the general rule is that an improvised weapon is an object, there is sufficient room to rule that this thrown-halfling attack by the half-orc fits the rule well enough.
Discuss this with your DM on that basis to see if the DM objects. I doubt many DM's would.
Improvised Weapons
... An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin. {snip}
An object that bears no resemblance to a weapon deals 1d4 damage (the DM assigns a damage type appropriate to the object). If a character uses a ranged weapon to make a melee attack, or throws a melee weapon that does not have the thrown property, it also deals 1d4 damage.
An improvised thrown weapon has a normal range of 20 feet and a long range of 60 feet. (From Basic Rules; p. 47)
If you were thrown more than 20 feet at the target, the attack is made with disadvantage. (Roll 2 d20's, take the lowest result).
You do your monk attack upon arrival, as you see fit and as initiative order allows. Your DM may want to limit you to a single attack (like a Ready action) based on how you described "lining up" your punch. A generous DM may allow you to do a full monk attack routine upon arrival. That's within DM discretion to decide.
RAF. Regardless of what’s on the page or what the designers intended, D&D is meant to be fun, and the DM is the ringmaster at each game table. The best DMs shape the game on the fly to bring the most delight to his or her players. Such DMs aim for RAF, “rules as fun.” We expect DMs to depart from the rules when running a particular campaign or when seeking the greatest happiness for a certain group of players.
For more insights on using and abusing small-sized humanoids, or even enemies, as improvised weapons, we have Q&A for how to "drop kick a gnome" and how to "use an enemy as a weapon."
While popular in fiction like comic books and anime, there are no published rules for one character throwing another.
While D&D is not a physics simulator, effective thrown weapons are some combination of small, light, and designed to purpose (i.e. to be thrown). A humanoid body is none of these. It's a lumpy, asymmetrical, jointed bag of mostly water, and has no decent aerodynamic properties.
For a Monk, Martial Arts and Unarmed strike lets you use any body part as a weapon, thus you can kick instead of punch.
I would allow this and the following is how I see it.
The orc expends 5 ft movement to make a spin/discus style throw. You convert your flight into a flying/jump kick and strike the target in the chest/head; using your legs to absorb impact and convert momentum into added damage. You possibly knock the target prone and land on your feet in front of the target.
The orc rolls to hit using STR bonus; 20 Ft normal, 40 FT long range. A failure means you land in a random square next to the target. If that square is occupied by an enemy, you may switch target to that enemy and proceed as if the roll was a success. If the orc rolls a natural 20, then you get +2 on your Athletics(DEX) check.
You roll an Athleltics(DEX) check to see if you successfully convert your flight into a kick. A fail means you take 1d4 impact damage. If the orc succeeded in his attack roll, the target also takes 1d4 damage. If the throw is 20 ft or less add the orc's STR bonus to the damage. If you roll a natural 20 on your check, consider your attack to be a critical hit (double damage).
If both rolls succeed, then you roll normal damage for unarmed strike (1d4+DEX or STR). If the throw is 20 Ft or less, add the orc's STR bonus to damage and the target needs to make a STR save or be knocked prone. You can add Ki Empowered Strike or Stunning strike to the attack for the cost of 1 Ki.
Note: A flying kick only allows one hit if you wish to land on your feet, so Flurry of Blows cannot be added to the attack.
Since you are making an attack outside your turn, I'd rule that the orc has used an action, and you have used a reaction.
