No it doesn't work.

I just tested it fairly extensively in a custom game.
Lux's snare works by itself since the DOT happens while the target is snared. So 1.5 secs of 5% damage.

Her circle doesn't get the bonus because the target is not impaired while the DOT is in effect. So 1.5 secs of 2.5% damage.

You could detonate it right after they get snared for the double-half damage, but that doesn't actually help much since it only refreshes the proc from the snare and once the snare ends, it's back to half damage anyways.

That's why Rylai's has good synergy with Liandry's - the DOT and slow are happening together, so it gets the damage bonus.

Wait, what? Why would the target have to be snared for the whole duration?

It's an on-damage effect, a debuff.

As long as the target is snared, any magic damage done by the champ with the item will apply the burning debuff that won't change while it's running. Debuffs never dynamically change while they're in effect. They have to be reapplied for that.

I don't understand what you're confused about.

The DOT does double damage as long as the target is movement-impaired. It does normal damage if the target isn't.
So if the you pop Lux's circle after they're snared, once the snare times out, the DOT will do normal damage. Before the snare times out, the DOT does double damage. It does change while it's running - not really that hard to imagine that the mechanics of it is that it checks whether or not the target is movement-impaired for each tick of damage.

Also, the debuff is only applied for spell damage and not just any magic damage.

You don't need to slow them for the snare effect to do double damage. A snare affects movement, and the item description is "if their movement is impaired," they take 10% of their current health in bonus damage. I think the original question might have been, "If I impair movement multiple ways (i.e. snare and slow) does the 10% double?" In which case the answer is no. Alternatively the question might have been "does lux's e proc Liandry's torment passive," in which case the answer is no, because their movement is no longer impaired. Last, the question could have been, "If I snare someone using my q, then detonate e while they are still impaired by the snare, does it proc liandry's torment's passive?" In which case the answer would be "yes," because their movement is impaired. The question could have been phrased much better since her e is not a snare, and the question begins "if I've snared someone w/ my e."

If counter to all my english skills you are trying to actually ask "how do I maximize the damage from my kit using liandry's torment," see Beaumains's post, as it is rather informative.

DoT spells halve the duration. Lux' E is a slow while undetonated. So you'd get 10% for half the time while the E is undetonated (5 seconds). Of course, they could just walk out of the E and take damage for even less time, but it SHOULD work on an undetonated E.

But that doesn't make any sense...

Name one other such non-channeled DoT debuff in the game that dynamically changes damage after being applied.

This isn't a channeled spell like Fiddlesticks' Drain.

It's a singular debuff that's applied to an enemy target.
If it's applied while the enemy is movement impaired, it'll do more damage for the entire duration of the debuff.
If it's applied while the enemy is NOT movement impaired, it'll do less for the entire duration.

It can't change suddenly after being applied.

If you don't believe me, test it yourself.
I'm just reporting what I saw when I tested it.

The only debuff that depends on the state of the target is Viktor's Gravity Field, which requires the target to already be slowed to build another stack up to ultimately getting stunned. That's why Yi, say, can run through without getting stunned. Apart from that, there are plenty of debuff effects that decay or build over time, like Lulu's Glitterlance and Trundle's Agony. So dynamic debuffs definitely exist, and debuffs that have to check for the target's current state also exist.

On the other hand, we don't have any previous examples of a DOT that does double damage under some condition at all. So why be surprised that the first one that gets revealed has a dynamic component? Isn't that better and more intuitive given that the description says "if their movement is impaired, they take double damage from this effect" and not "if they're movement-impaired on cast, they will take double damage for the full duration"?

I think that you're imagining two versions of the debuff - a double strength one and a normal one. What's actually the case is that there's only one version that checks for the movement-impaired condition for each tick of damage.

Those aren't dynamic debuffs though.

When Glitterlance hits a target, it applies a debuff that slows the target with a slow that decays over time.
The slow works that way, by decaying. It's still a static slow effect though.
Nothing can make that debuff different after it's been applied. Absolutely nothing. It will work exactly as it's supposed to (slowing for 80%, then decaying), and there's no way to change that behavior after it's been applied.

Non-terrain debuffs don't change after being applied.
Viktor's W is a terrain based spell, so that's different.

Take, for instance, debuffs and abilities that do extra damage to targets below a certain % of health.
Once you apply the debuff or ability to a target below that % health, it'll do the full amplified amount of damage even if the target goes above that % of health while the debuff/ability is afflicting them.

This discussion is pointless.
If you don't want to accept that the DOT's damage changes dynamically, that's fine with me.

If you see it as something new, then it's something new. I don't understand why it bothers you so much.
I don't care enough to keep trying to convince you though.

Why not?

Why is Riot not allowed to add in functionality for dynamic buffs and debuffs?