Being dropped is not movement, but check with your DM
Movement is a defined term for creatures (p. 190, PHB)
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here. Your movement can include jumping, climbing, and swimming.
If the creature is not spending any of its speed, it is not moving. It maybe is being moved, e.g. from the effect of a thunderwave spell, or by thorn whip, but being moved is not moving. These spells also do not say they move the creature, thorn whip says "you pull the creature closer", thunderwave says it the creature is "pushed (...) away". This is similar to teleporting into the spike growth area, which also is not moving.
The rules for opportunity attacks also support the idea that only active movement by spending speed counts as moving for effects that harm the creature:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach. [...] You also don’t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction, For example, you don’t provoke an opportunity attack if an explosion hurls you out of a foe’s reach or if gravity causes you to fall past an enemy.
Therefore, the creature takes no damage from spike growth when being dragged around or pushed around by these spells.
There are two spells that could be used to argue that falling counts as moving, and that the stone spike talks about moving not in the game term sense, but in the more general, common-language sense of the word that means changing your position.
The first is Booming Blade, which demands that a creature moves willingly to suffer its effect. As you only can spend speed when you willingly do so, that addition would be unneccesary if to move always meant spending speed. However, this spell was published much later, and it is possible that the willingly was added to make it easier for readers to understand the intent.
The second is Antipathy/Sympathy, which says
he creature must succeed on a Wisdom saving throw or use its movement on each of its turns to enter the area or move within reach of the target
Again, if moving always would require movment, then stating the creature must use its movement to do so would be superfluous. Again, this is not clear cut, because the creature might also have other, better ways to get closer to the target, like teleporting to it, and maybe the text is there to make it clear so the spell does not force it to do this.
Lastly there is also the narrative perspective. While technically, spells only do what they say they do, some DMs consider what makes sense from a narrative perspective. If there is an area full of sharp spikes, being dropped onto them would make sense to cause damage.
So check with your DM, if they allow you to deal damage, or not.