If I have out a Karn liberated I exile some stuff, it leaves play and I restart the game later on with a different karn will I get stuff I exiled with other karns or is it specific to the karn that resets?
Where Karn Liberated says "all non-Aura permanent cards exiled with Karn Liberated", it refers only to all non-Aura permanent cards in exile that were put there due to that Karn Liberated's first two abilities (C.R. 607.1a, 607.4), not to cards exiled due to any other ability, not even an ability of another Karn Liberated (especially any previous Karn Liberated).
That means, if you restart the game due to Karn Liberated's last ability, only non-Aura permanent cards exiled with that Karn Liberated are put onto the battlefield when the game is restarted. All other cards in exile (even if they were exiled with another Karn Liberated), are not put onto the battlefield this way.
BWTeysa, Orzhov Scion Combo
GUEzuri, Claw of progress Morph
GUBSidisi, Brood tyrant
RWGisela, Blade of Goldnight Random red white cards i dont use.dec
GBLoam Pox
Modern
UBFaeries
GBWGoyfless Abzan
On Squirrels
On Risen Executioner
That means, if you restart the game due to Karn Liberated's last ability, only non-Aura permanent cards exiled with that Karn Liberated are put onto the battlefield when the game is restarted. All other cards in exile (even if they were exiled with another Karn Liberated), are not put onto the battlefield this way.