The damage type that Kayn deals is mostly Physical Damage.
Kayn's damage type breakdowns as follows:
At one point in the game, Kayn will permanently transform into either a Shadow Assassin or a Darkin.Shadow Assassin: Whenever Kayn deals damage against enemy champions he also deals 15% - 45% (based on level) of post-mitigation damage as bonus magic damage, lasting for 3 seconds upon entering in combat with an enemy champion, refreshing after 8 seconds out of combat with them or by using Umbral Trespass Umbral Trespass.Darkin: Darkin heals for 20% - 30% (based on level) of the post-mitigation damage that he deals with spells against enemy champions.
Kayn dashes forward to strike nearby enemies. Both dash and strike deal 65/85/105/125/145 (+80% bonus Attack Damage) physical damage.
Kayn deals 90/135/180/225/270 (+130% bonus Attack Damage) physical damage and slows enemies hit by 90%, decaying over the next 1.5 seconds.
Kayn gains 40% movement speed and the ability to walk through terrain for 7/7.5/8/8.5/9 seconds. When he first enters terrain, he is healed for 90/100/110/120/130 (+45% bonus Attack Damage).
Kayn infests an enemy champion he has recently damaged, becoming untargetable for up to 2.5 seconds. Reactivate to end this effect early. When Umbral Trespass ends, the target takes 150/250/350 (+175% bonus Attack Damage) physical damage.
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