All the material for Kayn shows him picking on our poor Frost Archer (his Shadow Assassin splash art, multiple moments in his Champion Spotlight), which makes me wonder if he might get a mission in-lore to try and assassinate her. Obviously, Riot would have him fail or scrub the mission, but at least it'd get things rolling in any sort of upcoming Freljord update.
Just some spitballing here.
I highly doubt he would try to assassinate Ashe. Why would the order of shadow possibly take interest in a political leader active in the opposite end of Valoran?
The only way i can realistically see the two crossing paths is if Rhaast somehow links Kayn up with Aatrox, and Ashe for some reason accompanies Tryndamere in his quest for vengeance against the Darkin of War.
They only depicted Shadow Assassin Kayn attacking Ashe because SA Kayn works best against squishies, and Darkin Kayn is depicted attacking Sion and Darius because he's a tank killer with his percent health damage.
