Would a frightened, fleeing creature be considered moving willingly?

Question

If I were to cast a spell on a creature that causes it to become frightened of me and spend all of its movement getting away from me, would that count as "moving willingly" for the purpose of triggering effects like Booming Blade's secondary feature?

This could be viewed from two angles. Either, you are forcing the creature to move, rendering its movement "unwilling" - or you have frightened the creature so much that it wants to get away from you, thereby rendering the movement "willing", to some extent.

Note that, while they also don't trigger on forced movement, opportunity attacks still trigger on frightened, fleeing creatures (whether willing or not), since the description specifically states "You also don't provoke an opportunity attack when [...] something moves you without using your movement, action or reaction".

In this question, which was about 4e (!), the consensus seems to be that running away due to a magical fear effect would not be willing. However, I don't see what would be the logic behind Booming Blade taking into account that a creature moves out of fear. Does the spell "decide" "Hey, this guy wouldn't have moved if he hadn't been enchanted. Let's not deal damage"? Hardly.

Spells that could be part of a situation where this is relevant:

Booming Blade:

[...] If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends. [...]

Fear:

[...] While frightened by this spell a creature must take the Dash action and move away from you by the safest available route on each of its turns unless there is nowhere to move. [...]

Answer 1

Unwilling Movement won't trigger Booming Blade

Fear provides the following direction for those under it's effects, emphasis mine(PHB, 239)

While frightened by this spell, a creature must take the Dash action and move away from you by the safest available route on each of its turns, unless there is nowhere to move

And Booming Blade requires, emphasis mine (SCAG, 142)

If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.

While the creature is moving of their own volition, the movement itself is not willing. This would trigger an Opportunity Attack (they aren't physically moved like through Thunderwave, but are moving on their own unwillingly), but because the movement is unwilling, Booming Blade's secondary feature would not be triggered as they are not moving away willingly.

Magical Persuasion does not translate to Consenting Decisions

The crux of this is that a creature is not willing if the decision they make was not internal. Either through something like Fear, Suggestion, or Charm, a creature under those effects has not made a willing decision. An outside party has decided for them. They may think it's on their own, but it's not. You can't consent to something when you aren't in control of yourself to have that consent.

Answer 2

A frightened creature spends remaining speed, which constitutes willing movement

Apparently, the "willing/unwilling" distinction is never defined properly in the rules. So, any adjudication of this problem will have to rely on speculation based on common assumptions about the nature of will.

Before diving into the details, it is worth pointing out that this problem is confounded by equivocation between a player's and a PC's will. In some sense, a PC doesn't have a will, because it is a figment of imagination. However, for the sake of the game, we pretend that the PC is capable of will. While the spell puts limitations on the player's will, it does rather the opposite for the pretend will of the PC. It exaggerates or amplifies the PC's will. The fear of the frightened PC informs its motivations; it flees because it wants (read: wills) to.

Nuance:

The fact that they can use the

safest available route

means that they have at least some exercise of will. I.e. they can willing choose to avoid danger. So, even though the real world player cannot willingly control the PC, the frightened PC willingly chooses to flee (so long as it can). And, reasonably, the frightened creature wants to flee. I.E. It is willing to flee, even if "there is nowhere to move".

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Interpretations:

Conservatively:

If we say that the fear is so overwhelming that the creature is coerced into movement, and that something like "pure, tranquil consent" is necessary to qualify as "willing movement". We can make a (rather weak) argument that the movement is unwilling. But, this reasoning hinges on semantic nuance that strictly ignores the desire (read: will) of a frightened creature, to flee along the safest route.

Liberally:

There aren't proper rules to cite, but for practical purposes, the line between willing and unwilling movement can be neatly drawn where Willing movement costs remaining speed; unwilling movement is caused by some external force or trigger, and it does NOT consume remaining speed. Accordingly, in this case, Booming Blade should trigger.