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Wikipedia Article:
The original Exiles team consists of Blink (Clarice Ferguson, from the Age of Apocalypse continuity), Mimic (Calvin Rankin), Magnus (son of Magneto and Rogue), Thunderbird (John Proudstar), Nocturne (Talia Josephine "T.J." Wagner, daughter of Nightcrawler and Scarlet Witch) and Morph.
When Magnus dies (Exiles #2), he is replaced by Sunfire (Mariko Yashida). Sasquatch (Heather Hudson) arrives when Thunderbird is made comatose during a battle with the world-eating Galactus (Exiles #10).
In Exiles #12, the Exiles discover there is a second team conscripted by the Timebroker, Weapon X, whose missions typically involve killing or maiming innocents and heroes. The two teams join forces to rescue a group of children from a Sentinel prison camp. However, the teams are told the second stage of the mission is to kill one of the children - David Richards (son of Rachel Summers and Franklin Richards) - who will apparently grow up to become a powerful supervillain. Weapon X are willing to kill him, but the Exiles are not and the two teams fight until the Timebroker intervenes. Sabretooth (of the Weapon X team) agrees to stay behind and raise David Richards himself.
The Timebroker sends Blink home after a visit to a world plagued by a variant of the Legacy Virus, since the team had repaired her personal broken chain in time (Exiles #22) and Magik (Illyana Rasputin) takes her place. Mimic then replaces Blink as leader, gaining the Tallus.
During Exiles #28-30, the Exiles arrive in the main Marvel Universe timeline, Earth-616, where they meet the X-Men after Havok's cancelled wedding in Uncanny X-Men #425-426. The Exiles team up with the X-Men against an evil Havok from the Mutant X universe, who shares a body with the good Havok. After Havok is subdued, the Timebroker arrives to personally eliminate the Mutant X Havok's consciousness.
After Sunfire is killed by a Brood-infected Mimic (Exiles #37), she is replaced by the Exiles' former teammate Blink. During the next mission (Exiles #43), the Exiles and Weapon X are forced to battle each other until only six remain alive. Magik is killed by Hyperion when she attempts to switch sides and betray the Exiles. Hyperion is eventually defeated by Blink, who teleports his own heat vision into his back, paralysing him. Gambit (of Weapon X) is allowed to deliver the killing blow against Hyperion with Magik's sword, which actually kills them both, leaving five Exiles alive and their mission complete.
The Exiles visit the main Marvel Universe, Earth-616, for a second time in Exiles #46-48. They are joined by Namora, who replaces Magik, and are instructed to "leave their possessions and earn their wings". Reed Richards deduces this means they are to leave Nocturne behind (whose power involves possession) and have Beak join the Exiles, which is confirmed by the Timebroker.
In Exiles #53, the team are told by The Celestials to "beware the Timebreaker, he is not what he seems". After this, the Timebroker's behaviour becomes increasingly strange. Heather Hudson is removed from the team (Exiles #59) without explanation and replaced with former Weapon X member Sabretooth. The tallus orders the Exiles to kill Mimic and despite their refusal, they are allowed to move onto a new mission (normally a mission must be completed for the Exiles to move on). Next, the Timebroker replaces Beak with Holocaust (Exiles #60), which they are informed is a punishment for disobeying previous orders.
Eventually, the Exiles break free of the Timebroker and stage a raid on Panoptichron (also known as the "Crystal Palace"😉, home of the Timebroker and a location from which many parallel universes can be monitored. Here they discover that the Timebreakers are an alien race of bug-like beings who found the Panoptichron and accidentally broke a series of timelines. Lacking the power to repair the timelines, they concocted this scheme of recruiting heroes from various worlds to do their work for them. Not only that, all previously "returned" heroes are actually in Panoptichron, frozen in blocks, along with the killed heroes. As a final twist, the evil Hyperion has broken free of his prison and has taken over Panoptichron. The Exiles defeat Hyperion, but at the cost of Namora and Holocaust's lives. Beak returns from stasis and saved the day by calling on two good versions of Hyperion for help, fulfilling his destiny as an Exile. Only Blink, Mimic, Morph, Sabretooth, Beak, and Heather Hudson remain to pick up the pieces.
The Exiles then return to Earth-616 to take Beak home. Unfortunately, Earth-616 is dominated by the House of M. During their stay, the Exiles run afoul of the body-hopping serial killer Proteus, who steals data from Panoptichron and escapes from reality to reality after the House of M ends, leaving Beak depowered and back home. The Exiles chase Proteus through several realities, including New Universe, 2099, Squadron Supreme, Future Imperfect, and Heroes Reborn, before finally bringing Proteus down by trapping him in Morph’s body, which does not decay, unlike all of Proteus’ other host bodies. During the “World Tour” of hunting down Proteus, Mimic is possessed by Proteus and dies, the Exiles create divergent timelines in 2099 and the New Universe, and pick up new members Longshot, Spider-Man 2099, and Power Princess.
The Exiles begin cleaning out the stasis gallery of former Exiles and Weapon X members and send them back home. Iron Man, Daredevil, and Angel are sent back to their respective realities alive, while every other Earth with a missing superhuman holds a funeral. Blink, at the suggestion of Power Princess, takes Mimic home to be buried with his X-Men instead of burying him in Panoptichron. Spider-Man, Sabretooth, and Heather Hudson all decided to visit their home realities. Unfortunately, the Timebreakers ditch the Exiles when the Timebreakers believe that they are not saving realities as they should be. However, after using several squads of Wolverines to complete the next mission and failing, Logan (from Days of Future Past) and young James Howlett convince the Timebreakers that the Exiles are needed. The Exiles finally resume their reality-saving missions for the first time since they discovered Panoptichron when they save a world being destroyed by the Silver Surfer.
Then, Heather discovers an Earth inhabited by a disturbingly close approximation of the original Exiles team, complete with their own Timebroker. After the "classic Exiles" fight "the all-new Exiles" (as Morph put it), it is later revealed that the whole thing was a scheme orchestrated by the Grandmaster, who seeks revenge on the Exiles for freeing Professor X on their first mission and ruining a bet. The Exiles, past and present, join forces with the Wrecking Crew of that Earth to defeat the Grandmaster. The current team then leaves the Earth in the hands of the approximation Exiles. The Exiles then take three weeks to save a chain of cracked realities, injuring Blink, Morph, and Spider-Man 2099. They also discover that Proteus possibly immune to metal while in Morph’s body. Meanwhile, former Exile Thunderbird is in the stasis gallery, dreaming of what might have been had he not become comatose.