I know that infestation movement does not provoke opportunity attacks. However, I have a different question:
Does the movement caused by the infestation spell trigger the extra damage of booming blade?
The description of booming blade (SCAG p. 142) states:
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.
And infestation (XGtE p. 158) says:
You cause a cloud of mites, fleas, and other parasites to appear momentarily on one creature you can see within range. The target must succeed on a Constitution saving throw, or it takes 1d6 poison damage and moves 5 feet in a random direction if it can move and its speed is at least 5 feet. Roll a d4 for the direction: 1, north; 2, south; 3, east; or 4, west. This movement doesn’t provoke opportunity attacks, and if the direction rolled is blocked, the target doesn’t move.
Best Answer
No. Movement forced by a spell is not willing movement.
"Willing movement" (and being "willing" in general) is never defined in the rules, though it's referenced in spells like booming blade (SCAG, p. 142). Thus, since it's not a defined game term, we default to the natural English definition of the term.
According to Dictionary.com, "willing" means "disposed or consenting; inclined". However, the movement from the infestation spell (XGtE, p. 158) is forced by the spell on a failed Con save; the target has no choice about whether to do it, so it's certainly not doing so of its own volition. As such, it seems clear that movement forced by such a spell is not willing.
Jeremy Crawford supports this interpretation in an unofficial April 2016 tweet in which he addresses a similar question about the dissonant whispers spell and its interaction with booming blade:
Crawford provides the same explanation in an earlier unofficial tweet in February 2016:
The same logic would follow here regarding infestation and booming blade; the spell is forcing the targeted creature to move, so the creature is not doing so willingly. Thus, booming blade's extra damage would not trigger.
(Sidenote: infestation does not trigger opportunity attacks either, because it moves the creature without using its action, reaction, or movement.)