Red Kayn is far easier to deal with than Blue.
Grievous wounds, sidestep E and CC. If he used Wall Walk to get near you, he won't have it for awhile. That's an opportunity to poke. CC composition. (Poppy's really fun because if he tries to wall walk, she can ram him into it)
Seriously. Most of his power is in his E. If he invades you, depending on the champion, you have time to react. Save your Flash for as soon as he emerges, because he will try to finish the job. You may survive with 20-30% HP if you do, and either trade him back or escape. Barriers work well, as does Zhonya's.
Blue is far more insidious, because raw 30% Magic damage added and his speed is toxic. And he usually is ahead if he chooses it.
Just had a "Plat Orianna" with Victorious skin..... in a Silver game..... yolo push mid turret, and int feed Akali/Kayn while doing it. Sacrificed 7 deaths just to get the turret.
Kayn ofc then went Blue, and could literally 1 shot anyone. Literally had to outplay him to kill him. We had a Kai Sa adc, who sucked early game. But....late game she popped off and her R's repositioning allowed her to kill him a few times, simply because it was a case of burst vs burst, and she had better mobility he couldn't track. But as Nami I couldn't do crap. I knew if he burrowed a wall near me I was dead. I started to build either a Randuin's or a Thornmail, and on the last skirmish I lived by about 20% HP, but Yorick was backdooring at that point so it didn't matter (Team cared more about baron than defending a base).
Sucks. Because in that game I literally solo'd Jhin 6 times as Nami. I solo shut down a 4 man bot turret push too. Didn't matter, because I was Nami and Orianna fed an Akali and a Blue Kayn.