The Fifth Doctor:

Tegan, Adric and Nyssa stay with the Doctor as he gets used to his fifth regeneration.

Adric dies while attempting to defuse a cyber bomb in 'Earthshock'.
Tegan leaves at the end of 'Timeflight', but reappears in 'Arc of Infinity'. Many of her relatives turn up in the stories, all-be-it relatively briefly! She returns to her own time after the events in 'Resurrection of the Daleks'.
Nyssa leaves the TARDIS in order to help the Vanir synthesise the drug they require to live in 'Terminus'.
Turlough joins the Doctor during the events of 'Mawdryn Undead' where we also meet the retired Brigadier who is now teaching A-level maths to the Doctor's surprise. Turlough is actually an exiled alien having fought on the wrong side in his planet's civil war. He returns home after the rescue of his family from the prison planet of Sarn in 'The Planet of Fire'.
Kamelion was rescued from the hands of the Master after a failed attempt to stop the Magna Carta being signed in 'The King's Demon'. Kamelion had the ability of being able to change his shape in order to 'become' anything required. It's not quite clear when, or if, Kamelion ever left the TARDIS - maybe he's still stuck in one of the many rooms that make up the ship.
Perpugiliam (Peri) Brown joins the Doctor, Turlough and Kamelion after a holiday to her step-father's archaeological dig ended up with her mother and step-father dead. She actually lasts the longest of any of the Companions. Unfortunately, this was largely due to the series being 'rested' for a season.

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