Extradition, or extradition procedure, was the legal process of one government requesting the transfer of an individual from another government's jurisdiction to their custody for alleged crimes within their jurisdiction. Extradition without due process was prohibited under the terms of intergalactic treaty. (TOS: "Let That Be Your Last Battlefield")
When the Federation received an extradition request, a hearing was held to determine if enough evidence existed to warrant a trial. If it was decided that there was sufficient evidence, the charged individual was turned over to the other government and then was subject to their legal system. Establishment of motive, method, and opportunity were considered sufficient under Federation law to warrant extradition of the accused. The Bajorans handled extradition requests in a similar fashion. (TNG: "A Matter of Perspective"; DS9: "Dax")
In 2286, the Klingon Ambassador requested the extradition of James T. Kirk so that he could stand trial for the killing of a Klingon crew member, the theft of their ship, and the creation of the Genesis Device. The Federation President rejected the request, stating that Kirk would face trial in a Federation court for violating nine Starfleet regulations. (Star Trek IV: The Voyage Home)
William T. Riker was the subject of an extradition request by the Tanugan security force in 2366 when he was accused of the murder of Doctor Nel Apgar in orbit of Tanuga IV. Though initially it appeared as if Captain Jean-Luc Picard would have to grant extradition, due to the evidence, it was eventually found that the doctor had killed himself while attempting to kill Riker. The security force subsequently dropped its request. (TNG: "A Matter of Perspective")
Jadzia Dax was the subject of an extradition request in 2369 by the government of Klaestron IV, also for murder. This case was unique in that the alleged murder was committed by the previous host of the Dax symbiont, Curzon Dax. The request was also complicated by the fact that Deep Space 9 was a Bajoran facility, and the Klaestron government had no extradition treaty with Bajor. This required the Bajoran judge Els Renora to hold a hearing. Eventually, an alibi for Curzon was discovered, and the request was dropped. (DS9: "Dax")
The Klingon Empire requested the extradition of Worf in 2372 for the alleged murder of 441 Klingon civilians by destroying their unarmed Klingon civilian transport ship during a battle. The hearing, presided over by Admiral T'Lara, eventually found that the ship Worf ordered destroyed was empty, and the allegedly dead civilians had died in a separate accident. The Klingons had hoped to stop Federation aid shipments to the Cardassians due to embarrassment from the incident. (DS9: "Rules of Engagement")
While impersonating a member of Lurian Security in 2374, Hain claimed to have an extradition order for Quark. (DS9: "Who Mourns for Morn?")
In 3191, President T'Rina of Ni'Var claimed to Primarch Ruhn of the Breen Imperium that she needed to prepare extradition orders for Moll and L'ak before handing them over to him. T'Rina admitted that this was a lie, quipping that it was better than claiming a tribble infestation as a delaying tactic. (DIS: "Erigah")