A couple of ambiguities, and a serious balance warning
There's nothing that's been clearly defined in the Rules as Written that would prevent this strategy from working. If your DM is in favor of it, and lets it work, then have fun! That being said, there's several ambiguities in the rules that may complicate this strategy.
Before we get into them, though, there are a couple of rules that could actually be used in favor of this strategy, and could even improve it.
The good news (what works even better than you propose)
You probably won't fall prone:
One minor rule works in your favor (that you mentioned in your question) is that you likely won't fall prone after jumping 20 feet in the air. First of all, the rules on falling state (PHB, p. 183, bold added):
At the end of a fall, a creature takes 1d6 bludgeoning damage for every 10 feet it fell, to a maximum of 20d6. The creature lands prone, unless it avoids taking damage from the fall.
Since you definitely won't take damage from the 20 foot fall (due to Enhance Ability), you don't need to fall prone. And the fact that your opponent would fall prone if it took damage will not necessarily cause you to become prone. Jeremy Crawford has (unofficially) confirmed this on twitter.
Question: Say you grapple a target, and shove them prone. Does the grappler suffer any ill effects aside from action usage? RAW/RAI?
Jeremy Crawford: The rules make no distinction between grappling a prone creature and grappling a non-prone one.
This means that each leap will only take 5 fewer feet of movement than you previously proposed. So in your example strategy, you'd be able to jump (300/25/2) = 12 times, instead of your proposed 10.
You could do more damage by jumping less distance:
I wanted to add a quick note on optimizing this strategy (both as advice, and as a warning to demonstrate how this strategy could become unbalancing). You're currently using up a very important resource every turn when you use this strategy: your bonus-action. For a monk, this means you get two fewer attacks than usual. And for a level 18 monk (as you are proposing), that means a serious difference in damage.
Your proposed strategy would do 12d6 damage from falls, and damage from one or two attacks (two on subsequent turns, one on the first turn where you used an attack to grapple). So that would mean on average you'd do:
- 12d6 + (1d10 + 9) = 56.5 average damage first turn
- 12d6 + 2*(1d10+9) = 71 average damage subsequent turns
However, with a strength of 29 (belt of Storm Giant Strength), you'd be able to jump 12 feet up without Step of the Wind. To jump 10 feet straight up, you'd just need to use 15 feet of movement (5 to run, 10 to jump, and nothing else since you don't drop prone). This could be done without Dashing 150/15/2 = 5 times per round for your proposed character, and would permit you to use your action and bonus action for attacks as well (though one attack would need to be a grapple on the first turn)! So this would mean an average damage of:
- 5d6 + 3*(1d10 + 9) = 61 average damage first turn
- 5d6 + 4*(1d10 + 9) = 75.5 average damage subsequent turns
So a savvy player could leverage this strategy to transfer most of their movement into damage without losing much of the damage dealing potential of their action and bonus actions. In fact, their expected damage would considerably increase over non-grapplers, since you could attack your opponent while they are prone with advantage.
All of this assumes that everything in your plan works as intended. But as I mentioned before, there are a couple of wrinkles to this idea.
Descending from a vertical jump may not cause damage
It's possible that you don't take any damage from a fall if that fall is the result of a high jump (and doesn't exceed that jumps' height).
This issue is not clearly defined in the rules: it's up to a DM whether there is a difference between descending after a high jump and "falling." However, the (currently) highest upvoted answer on this issue (to the question "Do you take falling damage after a high jump of over 10 feet?") indicates that you wouldn't take damage from such a fall. (And unless I'm misreading it, there's nothing in the answer that says you can choose to take damage from a jump, as you parenthetically propose in your description of this answer).
This is also backed up by some (unofficial) designer commentary by Jeremy Crawford on Twitter:
Question: [Do] you take falling dmg from the high jump?
Jeremy Crawford: In such a circumstance, I'd consider a fall to be a drop that exceeds the distance of the jump.
You might argue that while you do not take damage from such a fall, your opponent would. This is possible, but relies on some fine distinctions, such as whether your opponent descent counts as a fall or a jump if they were not the one jumping. Such a distinction will need to be left up to the DM.
It may be harder to move a prone target
The rules for being prone state the following (PHB, p. 292):
A prone creature’s only movement option is to crawl,
unless it stands up and thereby ends the condition.
And crawling as defined elsewhere states (PHB, p. 191)
To move while prone, you must crawl or use magic such as teleportation. Every foot of movement while crawling costs 1 extra foot. Crawling 1 foot in difficult terrain, therefore, costs 3 feet of movement.
Now, the rules stated above are clearly intended to refer to the prone target's intentional movement. But a highly pedantic (but legitimate) reading of the rules above would conclude that a prone creature that is moving through nonmagical means is to be considered "crawling". And the statement "every foot of movement while crawling costs 1 extra foot" does not explicitly only refer to movement spent by the prone creature.
As such, a grapppler moving a prone creature may have to spend 2 feet of movement for every foot they move this grappled prone target. This is in addition to the fact that the grappler's speed is halved while they move a grappled creature.
Again, your DM may have to rule on this. There is some realism to this proposed mechanic: the fact that a prone body is much harder to move than an upright one was used extensively as part of "passive resistance" in several nonviolent-resistance movements (such as the Struggle for Indian Independence). But a DM will need to decide.
You may not be able to jump high while grappling, especially if you want to damage the target with a fall
The rules on grappling state the following (PHB, p. 195 bold added):
When you move, you can drag or carry the grappled creature with you...
Now, this typically means you can move your opponent wherever you want as long as you are also moving in the same way. However, we need to consider which of these terms actually applies to to your proposed jump.
It doesn't really make sense to say that you "drag" an opponent when you are jumping straight up with them, since they aren't in contact with anything to create resistance. However, "carry" could definitely apply, and would allow you to elevate an opponent by jumping. However, if you "carry" your opponent, it is possible that they would never make contact with the ground when you descended! You would make contact with the ground and take no damage (possibly because of Enhance Ability, or because you may take no damage from descending from a high jump). Then, since your opponent is being carried (and therefore lifted), they may be suspended above the ground at the point of impact, and thus take no damage from the fall.
Now of course, you don't have to be carrying your opponent! You could position it so that it is below you in the air (i.e. body slam it). However, such an action goes beyond simple movement. A DM may require you to use an unarmed attack to do such a thing, or even an Action (since it is not a well defined option in combat, it may fall under "Improvising an Action", PHB p. 193).
A DM may be hesitant to translate involuntary Movement into Damage
As has previously been pointed out, PCs optimized for speed can move a truly absurd number of feet per turn. As a simple example, consider the case where you chose to stack the above proposed spells with a single casting of Haste. This would permit you to double your speed, and also have an additional action used to Dash. Thus you could move [35+30+10]*2(boots)*2(Haste)*2(Haste Action spent dashing) = 600 feet per round (also known as 68 miles per hour).
If you attempted to use this proposed jumping strategy with such an absurd amount of movement, you could do a 10 foot jump 600/(15)/2 = 20 times every turn. This would result in an average damage of:
If you tried to maximize this for "burst" damage, you could play a Tabaxi character (already a popular choice for monks), and for a single round be able to move (30+30+10)*2(boots)*2(Haste)*2(Feline Agility)*4 (Using your Action, Bonus Action, and Hasted Action to Dash) = 2240 feet per round. Using this strategy and jumping 20 feet in the air 44 times (2240/25/2) you could do 88d6 = 308 average damage in a single round based on only your movement.
Compare this to the standard average damage of a monk with Storm Giant Strength with Haste cast upon them, which would be 5*(1d10+9)= 72.5 damage per turn (about 56% of the other strategy's damage). And that damage is mitigated by AC, which could mean that not all the attacks connect.
Now, your proposed strategy definitely has a high opportunity cost. It requires several concentration spells, and requires your character to be very high level. It also only works against one target at a time, and cannot be used on targets which are Huge or larger. However, it also provides a very large amount of damage (70 on average from falling alone) which has no saving throw and is not dependent on a target's AC. It essentially removes two of the most important defensive features in the game, resulting in a creature that resists damage almost exclusively through its Athletics or Acrobatics roll modifier (or immunity to being grappled).
By mostly or entirely bypassing a creature's defenses, your proposed strategy could be easily unbalancing. And since all the spells involved in this strategy are quite low level, you could potentially carry out this strategy every single combat (higher level spells such as True Polymorph could increase its effectiveness, by transforming you into creatures with higher base speeds).
All these features may come together to make a DM hesitant to allow you to carry out this strategy as proposed, which might make them look to the ambiguities in the rules mentioned above.
You should talk to your DM
Your proposed strategy as stated (without Haste and with 20 foot jumps) is not absurd or unbalanced. It does only slightly more damage than attacking normally, and requires considerable resources on the part of the party (low level spells, but at least one instance of Concentration). Your DM may be willing to let it work mainly because it's clearly tied to your enjoyment of this particular class: creating a character whose main power comes from their speed.
But if you intend to optimize this strategy down the line, you could tell your DM what you have in mind and see if they have any objections. We're all here to have fun, and any DM worth their salt will want that for their players. But if a strategy causes one class to double their damage output compared to other characters, that could cause sufficient balance issues and reduce the fun for the overall group.