When using mystic shot, liandry's torment does not proc. It works with the other spells, but not mystic shot. It could be the fact that mystic shot does physical damage, but liandry's says "spell damage" not "magic damage".

Repro Steps:
Pick Ezreal
Get Liandry's Torment
Use Mystic Shot
See no DoT

If it is only Magic damage, then the title should instead be "Liandry's Torment should say 'magic damage abilities' and not 'spell damage' in the tooltip." It's not just magic damage or abilities, it has to be both from what I've seen.

Mystic Shot triggers on-hit effects, not spell effects.