Zandronum

I'm attempting to make a multiplayer Sonic Heroes style mod, and so far my plan is to make functional classes with roles in that style.

Speed runs really fast but uses Doom's lesser weapons
Power is a slower guy but has high dps weapons.
Flight flys all the time, is capable of airlifting allies, has a slow firing plasma pistol with infinite ammo and the rest of his weapons are support abilities.

Specifically, I need to figure out how I can allow a flying player that can airlift allies.

You could try experimenting with inheriting from LoreShot, which "pulls" whatever you shoot at to you.

I successfully made a magical flying marine with the grapple shot. Obviously one thing that will need to be done is make sure it can pull allies as well, but otherwise both the big things with Flight are accomplished.

IIRC, loreshot projectiles affect allies too (although they do no damage as long as friendly fire is off).

Ahh, but damage is required to pull enemies, which means my grappling beam could have a damage effect. I made the shooter fire the grapple "balls" rapidly so that it could hold allies in midair, and that behavior makes it a worthy Chainsaw replacement for Flight (it can do the same with enemies but damage them in the process).