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The Spike Growth spell states that
When a creature moves into or within the area, it takes 2d4 piercing damage for every 5 feet it travels.
Say, the creature is also concentrating on a spell, and it is moved 10 feet through the Spike Growth by a Thorn Whip. How many Concentration saves does it make? In other words: How many instances of damage does it take (combined from Thorn Whip and Spike Growth)?
We can ignore a lot of the context here and boil down the situation to a more direct question:
Does spike growth deal 2d4 damage after each five feet of movement, or does it let you finish your movement, then count up how far you moved and roll a number of d4s equal to twice the number of five-foot increments you covered?
And the answer is: It's not clear, so it's up to the DM.
However, there are some considerations to take into account outside the rules as written. A ruling that is technically legal but is going to confuse the players is probably a bad ruling. So we should ask ourselves, if a creature is moving through a zone of spike growth, how do we expect that to work, as players at the table?
Do we expect a goblin to walk through the spike zone, emerge from the far side, and die instantly? Or do we expect the goblin to take damage as they move and eventually slump down, dead, in the midst of the brambles?
Or, more important to the players, if a player character is trying to run through the zone, do we expect to cross the whole thing, then suddenly die instantly as the DM rolls well on 12d4, dealing massive damage? Or do we expect the player character to fall unconscious halfway through and then be in an awkward spot as they make death saving throws while nobody can safely approach?
I would argue that in each case, the expectation of the players is going to be that they're taking small bits of damage with each step as they go along, not a lump-sum at the end, which can have very different effects. The most obvious interpretation is usually the most correct one. In this case: You take 2d4 damage after each 5 feet of movement that ends with you inside the zone.
So in the case you outlined, the caster would take three instances of damage, one from the whip and two from being pulled through the spikes, so they'd need to make three concentration checks in rapid succession in order to maintain the spell.
The rules for concentration checks state:
Taking damage. Whenever you take damage while you are concentrating on a spell, you must make a Constitution saving throw to maintain your concentration. The DC equals 10 or half the damage you take, whichever number is higher. If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
The caster concentrating on the spell is pulled via Thorn Whip for 10 feet inside the AoE of Spike Growth: hence the spell description says that the total damage is 4d4, i.e. 2d4 for every 5 feet. This is one source of damage, hence the caster has to roll only one concentration check.
Obviously, before taking the concentration check induced by passing through Spike Growth the caster has to make a concentration roll for the damage from Thorn Whip.
Let's suppose another scenario: Eddy the Mage is concentrating on a spell, facing two enemy druids, Edwin and Xan. They Ready an action: cast Thorn Whip as soon as Eddy casts another spell, trying to pull him inside the AoE of a Spike Growth. Suppose that both hit and pull Eddy inside the Spike Growth, Edwin for 10 feet and Xan for another 10 feet. In this case, there are 2 sources of damage, hence Eddy takes 4d4 piercing damage from Edwin's pulling and 4d4 piercing damage from Xan's pulling, resulting in rolling two saves. (Plus the direct damage from each of the two Thorn Whip hits.)
Another scenario considers a creature moving inside a Spike Growth breaking up their movement: let's say that they move 10 feet, take an action and then move another 15 feet, all the time inside the AoE of the spell. This corresponds to 4d4 piercing damage for the first movement, resulting in a concentration roll, and then 6d4 further piercing damage from the 2nd movement, for another concentration roll.
For more support to the above reading of the spell, let's think to a similar case, i.e. quickly walking on charcoals. While one is walking on them for some seconds, they do not feel much pain, but as soon as they stop the brain register the extreme heat and the damage. If they start again to walk/run, then the pain increases.
I see walking in the AoE of Spike Growth as a similar case: one can run among the spikes and thorns for some feet and stop: then, the pain kicks in, all the cuts and bruise start to give pain. This is the first amount of damage (and thus the first concentration saving throw). Walking again results in other cuts/bruises, hence in other damage and a further concentration saving throw.
Spike Growth does 2d4 damage every time you move 5 feet inside its area. It's a conditional trigger that doesn't care about whose turn it is- if its conditions are met, the damage is dealt.
There is no 'Move Action' in 5E, although you can use a Dash action to move up to your movement speed in addition to your normal speed, and the spell isn't interrupting anything - it's doing 2d4 damage every time you move five feet inside its area, because other things can happen while you're moving, including attacks, spells, and damage, and because spells do what they say they do, and nowhere in the spell text does it mention "at the end of their move action," which doesn't exist.
Thornwhip isn't a teleport effect, so there is no reason for the usual damage trigger (five feet of movement through the area of spike growth) to spontaneously become ten feet of movement and only trigger a single time.
All other spells who scale damage additively as opposed to separate triggers follow the clause of
IF Condition THEN "The damage increases by."
The lack of the explicit phrasing "the damage increases by" (2d4 for every additional five feet), is indicative of the fact that each trigger is a separate entity.
If a creature is pulled by a Thorn Whip into Spike Growth and is currently concentrating on a spell, it will need to do a separate save for each source of damage as per the rules on concentration:
(…) If you take damage from multiple sources, such as an arrow and a dragon’s breath, you make a separate saving throw for each source of damage.
In this example, Thorn Whip will be one source of damage and then Spike Growth seems to be a point of contention. The description states:
(…) [creature] takes 2d4 piercing damage for every 5 feet it travels.
This is somewhat ambiguous as it can either mean "the creature takes 2d4 damage after every 5 ft until it completes its movement" OR "the creature does all its movement first, then takes the cumulative damage as a single source". It is not possible to deduce from the wording alone, which interpretation was intended and a good argument can be made in favour of either, therefore your DM will have to make a ruling either way and stick to it for the remainder of the game.
The Spike Growth description can be read two ways (chunked or lump-sum), both reasonable parsing of the wording itself. Either way has RAW consequences that aren't narratively ideal for different cases.
There's no RAW justification for it, but one way to get the best of both narratively is to apply damage every 5ft for creatures that might not survive the full trip (if walking under their own power, they'll stop when they drop), but to lump damage together for one concentration save per continuous movement (not interrupted by taking an action). This would be a houserule, it doesn't follow from the rules wording.
5 feet is only special because of the way D&D handle distances in 5 ft units, not because of anything in the narrative, like spacing of actual spikes. You're getting scraped and pierced on the legs by multiple thorns as well as stepping on spikes every footfall. So mechanically, the game is bundling lots of tiny damage sources into chunks for the minimum unit of grid movement, or one lump sum for the whole movement.
If you think of the damage as mostly from every one or two steps (when you step on a thorn angled just the wrong way), not as your legs are moving through undergrowth, then separate damage every 5ft is about the right length scale, so it's actually not nonsense. Especially if moving slowly enough for each step to be a separate distraction, not blurred together into some amount of pain. (Assuming a medium or large creature with two or four legs, not a snake or ooze or wheeled creature, or a prone humanoid being dragged by a Thorn Whip... but it's just a game, we should really just relax... and in this case maybe just apply what the spell actually says.) So if you prefer not to houserule anything, you can still narratively explain what's going on with a concentration save every 5ft as a useful average.
Death saves: one damage event is one failed death save if already at 0, no matter how big (up to -max HP insta-death). If every HP of damage from many thorns was a separate event, even a 5ft drag through this would be 2 or more failed death saves.
Resistance rounding down: multiple smaller hits might be odd numbers, e.g. if you rolled 5 damage (the average on 2d4) for each of four 5ft squares, that's 20 total reduced to 10. Or it's 4x 5 reduced to 4x 2 = 8 total, in that best case where you round down 4 halves instead of none. Resistance to magical piercing damage not from an attack is somewhat rare outside of Barbarian rage.
Flat damage reductions: Heavy Armor Master only reduces damage from attacks; this isn't an attack. Otherwise it would be narratively appropriate that someone in plate armor (including metal boots) reduces the damage by 3 for every 5ft chunk. I don't know of other flat damage reductions: they're rare in 5e, along with damage thresholds.
Defensive effects that trigger or can be used on damage: Anything that can be used once (per turn or whatever) to mitigate a single instance of damage will be more useful on a lump sum. For example, just looking at Wizard abilities because I know there are some which cost your reaction to deal with one instance of damage:
An abjurer can project their Arcane Ward onto another creature within 30ft, having it take the damage instead. This is a reaction to another creature taking damage, so only works on one damage event.
Bladesinging (10th): Song of Defense - When you take damage, you can use your reaction to expend one spell slot and reduce that damage to you by an amount equal to five times the spell's slot level. If you're dragged through multiple squares, you can only reduce the damage from one of them. And if neither roll is above 5, you'll waste some even with a first level slot. Unlike Arcane Ward, any leftover isn't still usable on later rounds.
Order of Scribes (14th): One with the Word - if you take damage while your spellbook's mind is manifested, you can prevent all of that damage to you by using your reaction to dismiss the spectral mind, using its magic to save yourself. Again, a single damage event costs your whole reaction, and doesn't leave lasting protection for the rest of your turn.
Treating the damage as one lump sum for a continuous movement makes some sense for abilities like those, including from other classes, because they'd work on continuous damage like stomach acid.
There's no rule that ever allows lumping separate damage together for some purposes but not others, not that I know of.
Choosing based on narrative appropriateness for different situations is something you might do, if this detail ever mattered in a game and people at the table weren't happy with letting this be one more of the oddities created by the details of how any RPG abstracts reality to streamline gameplay. (Hit points in general are super weird if you think too hard about it, especially with full healing on a long rest.)
