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Transwarp beaming was an advanced transportation technique which made it possible to beam a person from one planet to an adjacent planet within the same star system or even, using advanced computations, onto a starship traveling at high warp. (TOS movie & novelization: Star Trek)
After Spock time-travelled from 2387 to the mid-23rd century, he found himself in an alternate reality and encountered Montgomery Scott, a time-altered version of the Scotty he was familiar with. Spock rationalized a decision to give Scotty the computational formula for an advanced transwarp beaming protocol onto a moving starship, in order to facilitate an alternate Kirk rejoining his ship. (TOS movie & novelization: Star Trek)
Starfleet confiscated Scott's formula for transwarp beaming. In 2259, John Harrison, a Section 31 agent gone rogue, used the formula and a portable transwarp beaming device to flee from his attack on Starfleet Headquarters in San Francisco by beaming all the way to Qo'noS, the capital of the Klingon Empire. Scott expressed great displeasure when he learned the formula had been abused by Harrison. (TOS movie: Into Darkness)