(covers information from several alternate timelines)
Transwarp beaming was a transwarp theory originally postulated by Starfleet officer Montgomery Scott, whereby it was possible to beam an object or person from one planet to an adjacent one, or even to a starship traveling at high warp.
In the alternate reality created by Nero's temporal incursion, Scott had a debate with an instructor of his, who maintained that transwarp beaming was impossible. In an attempt to prove his instructor wrong, Scott tested his theory on Admiral Archer's prized beagle, although the experiment was a failure and the dog was lost. Scott described the notion of transwarp beaming as being like "trying to hit a bullet with a smaller bullet, whilst wearing a blindfold, riding a horse."
In 2258, in the same alternate reality, while stationed on Delta Vega, Scott encountered Spock (who had traveled back in time from the year 2387) and James T. Kirk. There, Spock provided Scott with the equation for successfully achieving transwarp beaming, which, by the time Spock had traveled back in time, Scott had already perfected. Upon viewing the equation, Scott noted that it had never occurred to him to think of "space as the thing that was moving." With this formula, Scott was able to beam himself and Kirk from Delta Vega to the Enterprise, which was traveling at high warp towards the Laurentian system. Later, Scott was able to use this technique again, to beam Kirk and Spock's alternate reality-counterpart from the Enterprise, which was in orbit of Titan, to the Narada, which was orbiting Earth. (Star Trek)
|aperture • beaming • chamber • coil • computer • conduit • drive • factor • hub • network • probe • signal • signature • space • theory • threshold|
According to the novelization of Star Trek, in the alternate reality, Scotty had been toying with the notion of transwarp beaming for years. Spock revealed that the only thing missing from the theory was the correct field equation for the continuous recrystallization of dilithium while transport was in progress. This allowed the steady usage of power during transport. It was the discovery made by Scott in the future of the prime reality. After seeing Spock's complete theory, Scotty realized he could not resolve the central string as it had never occurred to him to look at the event from the perspective of the one who was beamed. Spock however stated it in fact did occur to Scott when he added the adjunct equations related to dilithium.
The fact that a transport pad was not present at the receiving end made beaming on target possible only with a 4 meter margin of error, as the destination co-ordinates were completely theoretical. Admiral Archer's beagle, after having traversed in subspace for over 6 months, was intercepted by the Enterprise while it was warping out of Earth orbit for the first mission under the command of Captain Kirk.
In The Truth About Tribbles, Part 1, set in 2259, Scotty successfully experiments with extreme long-range transwarp beaming. He sends his pet tribble back to Earth from the Iota Geminorum system by bouncing the beam off a couple relays along the way. In The Truth About Tribbles, Part 2, Scotty explains the intention was to allow Starfleet to beam living things between Earth and the Fleet stationed in the outer quadrants. It was also to test if it was possible to retrieve Admiral Archer's beagle back from space. At the end of the story Scotty accidentally presses a button on the transporter console and the beagle materializes onto the transporter pad.