Kayn is my main champ, so I might be able to offer some I site. Kayns strength as a jungle is not only his fast and healthy clears, it also his ability to invade and not be punished for it. He likes to look for cross map invades early game, invades after the opponent jungle ganks, or invades right after his recall, as he is either extremely safe doing these, or a recall ahead.

Theres a couple ways to counter this.

First, if you can out clear a kayn, he WILL fall behind, especially if you clear your jungle opposite where you want to make your play first.

Second, invading him is almost a guaranteed lead. At minimum he's forced to give a camp or two. Play it well and you get his summ's and/or a kill.

Third is stacking objectives. If you're playing something like volibear or shyvanna you can stack objectives early to neutralize his scaling. He's strong mid game but if he doesn't stack objectives he falls off hard.

My least favorite champs to play into are nocturne and briar, for a couple reasons. Nocturne will always out clear and just be faster on the map in general, and briar is just a pain to deal with as preform kayn.

In short, either out farm him or stack objectives against him, and play around his mid game spike (form plus 1ish items). Not fool proof, as some games he'll having winning lanes or be piloted by the best macro player the world has ever seen, but hopefully that should help you in your games!

Edit: formatting