Is a mounted creature unaffected by the secondary damage of the Booming Blade cantrip?

Question

I'm designing a boss fight for the first session of my next campaign. The boss in question is a mounted warrior; one of his abilities is from the Mounted Combatant feat (PHB, p. 168), which allows him to force an attack or spell made against his mount to target him instead.

One of my players' characters is a Paladin/Sorcerer, and his main form of attack according to him will be casting booming blade.

The description of the booming blade cantrip (SCAG, p. 142) says:

As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell’s range, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and it becomes sheathed in booming energy until the start of your next turn. If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.

Jeremy Crawford has unofficially clarified on Twitter that "When booming blade refers to moving, it means movement in the game's normal sense: moving X feet."

The mounted warrior will be directly controlling the mount:

The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge. A controlled mount can move and act even on the turn that you mount it.

So, RAW, it seems that if the mounted warrior is the target of booming blade and the attack hits, he will take the weapon damage (and at higher levels the additional thunder damage) - but he can freely direct his mount to move and carry him around the battlefield as normal, as he himself is not spending any movement, and therefore will not trigger the additional damage. This could be circumvented by targeting the mount, but in this case the mounted warrior is able to redirect attacks and spells onto himself as per the Mounted Combatant feat.

Am I interpreting this correctly, RAW?

Answer 1

RAW, it will not trigger the secondary damage

Simply because you are being moved by something else. When booming blade description mentions

If the target willingly moves before then, ...

you must be the source of the movement and you must be willing. You must be using your movement (up to your speed). Even movement from dissonant whisper does not trigger booming blade because you are not willing.

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It might be obvious, but if you hit the mount, the rider won't take the secondary damage when the mount moves, only the mount takes the damage.

Answer 2

My ruling: if the target is moved of his own will then YES it takes damage

I base this not around a strict reading of RAW, because frankly I can't find anything really clear either way, however:

In either case, if the mount provokes an opportunity attack while you’re on it, the attacker can target you or the mount.

This tells me that being on a mount breaks some of the basic movement rules, and sets a precedent that mounted movement can trigger at least one type of movement triggered action.

I would therefore rule that controlling your mount to move counts as a valid trigger for Booming Blade as well.

Addendum:

I am never stickler for RAW, and frankly would trigger it even if it wasn't willing. Getting on the horse itself tells me that the rider was willing for the horses movement to count as their own where it benefits them, and in fair trade the same goes when it doesn't benefit them.