They will fall if and when they deprive themselves of movement
At the start of the affected creature's next turn,
it must choose whether it gets a move, an action, or a bonus action; it gets only one of the three.
A mind-whipped creature that chooses to take an action or bonus action does not get a move.
A flying creature will fall if it is
deprived of the ability to move
So, the basic question here is whether "not getting a move" is the same as "depriving yourself of the ability to move". What do the flying rules mean by "ability to move"? Having a move, where this is a rules-defined part of your turn (Action, Bonus Action, Move), or being able to move at all - that is, not being paralyzed?
It is helpful here to look at the consequences of the Stunned condition.
A stunned creature is incapacitated [can't take actions or reactions], can't move, and can speak only falteringly
Note that a stunned creature can speak (albeit only falteringly) - that is, they can move their jaw, their tongue, their lips, and consciously control their breath. And yet they "can't move". So it is clear that when stunned, the "can't move" does not apply to the natural English meaning of move (which includes the ability to move parts of one's body). Rather, "can't move" means they can't take their move in the game sense, they can't change their position on the grid (and compare this with the Paralyzed and Unconscious conditions, where the creature "can't move or speak" and the intent is that the creature does not have conscious control of its body).
This understanding is further confirmed by the PHB errata, which say (emphasis mine):
Is standing up from prone considered moving? Standing up costs movement but moves you nowhere. When the game refers to you moving, it means moving some distance. It doesn’t mean making a gesture or standing up in place. To move while prone, you crawl or use magic (PH, 191).
Thus, for a flying creature to be "deprived of the ability to move" does not require that it be paralyzed or unable to move in a natural English sense, but rather will apply any time it has lost the ability to translate its Speed into Movement. And this is precisely what the mind whip spell does by removing a creature's move.
A mind-whipped creature will remain aloft up until the moment of its choice, but once it has chosen an action or bonus action, it has removed its ability to move for the rest of the turn and will then fall.
Narratively, having been subjected to the mind whip, it is so confused that it cannot do more than one thing at a time, such as 'walk and chew gum' (which would be a move and an action). By concentrating on its action (or bonus action) in its confused state, it cannot pay enough attention to moving so as to maintain itself flying.
Longer Answer and Response to Critiques:
Flying and the nature of Free Will
In his answer, Thomas Markov argues that "Because they are free to make that choice, to move or not to move, we cannot say that they have been “deprived of the ability to move". A creature has only been deprived of the ability to move when they cannot choose to move, or could not have chosen to move."
I agree with some of this, but it needs to be carefully applied over the course of the mind-whipped flyer's turn. At the start of their turn, they have not made a choice, have not been deprived of movement, and are not falling. They are, as he says, "free to make that choice". However, at precisely the instant in their turn at which they take an action, bonus action, or begin to move, they have then made that choice - and they cannot rescind it. If the flying creature does not choose to move, they have then removed that as a choice for themselves for the remainder of their turn - they do not get a move. As he says, "A creature has only been deprived of the ability to move when they cannot choose to move" - and once they have made the choice not to, they cannot then later choose to move. They have, in other words, deprived themselves of the ability to move for the remainder of the turn, and will begin to fall.
Thomas Markov also notes that any creature, even in the absence of being mind whipped, that expends all of its movement on its turn is at that point unable to move further. Through its own choice, it has become unable to move (further). He argues that if this election counted as depriving itself of the ability to move, then any flying creatures which took their full movement would fall at the end of their turns, which would be silly. This argument merits more serious consideration.
I have to admit that linguistically, there is a point to be made here. If I contend that when a mind-whipped creature chooses to take an action, it deprives itself of the ability to move for the rest of the turn, then why is it not the same thing for any flying creature to take their full movement, and thus deprive themselves of the ability to move for the rest of the turn? Why don't all flying creatures fall at the end of their move?
In order to answer, I have to go beyond the actual text and talk about RAI and the theory of a game round. In doing so, I recognize that I am going beyond what I can actually defend RAW and instead expressing my opinion. Ultimately this may be an unsatisfying explanation for many.
It seems to me that, RAI, the mind whip spell is about reducing the options for the target's actions. The intent is that a failed save forces the target to prioritize one part of its turn over the others. A target on the ground that chose to act would lose the ability to move. It does not seem like the intent of the spell to say that a target in the air has more options than a target on the ground, that on a failed save they can not only take an action but can also use their movement to remain aloft as long as they don't actually move in any direction. It does not make sense to me to say that if they had no movement they would fall, but if they have movement they cannot use, they can remain aloft.
Further, I believe that the intent of initiative and the combat round are to fairly adjudicate events that are actually happening simultaneously within the narrative. Even though each creature's turn is six seconds, the entire combat round itself with all turns together also totals six seconds. That is, on your turn you can do about six seconds worth of things, but these six seconds are occurring simultaneously with everyone else's actions during the same time period. The purpose of initiative is to help resolve any conflicting events within the overlapping time period, but it does not establish that all of your actions actually happen before all of the actions of the next creature in the initiative order.
Within this context, a flyer that is mind whipped and then chooses an action will not be capable of moving for the next six seconds - within six seconds of its own time, being unable to move, it will fall. However, a flyer that has used all of its movement during its turn has moved for its entire six seconds and then its position is in some sense 'frozen in time' - it counts as aloft for the purposes of other creatures' actions until the beginning of its next turn when its movement resumes. Simply using all the movement you have been allotted within one time period does not make you fall, because you never actually stopped moving - there is a seamless transition from the movement you made on your last turn to the movement you are still making at the start of your next turn, regardless of the language of the movement rules. A mind whipped creature that can't move, in contrast, does not get to move on its entire turn and is treated as such until the start of its next turn.
It is clearly silly to contend that flying creatures fall at the end of each turn, but I find that Thomas Markov's argument of 'you can't ever deprive yourself of something as long as it was a free choice' to be unsatisfying as well.
First, because in English "Depriving oneself" is a standard expression, and can be found in many quotes, including one attributed to Franklin Roosevelt:
Nobody will ever deprive the American people of the right to vote except the American people themselves and the only way they could do this is by not voting.
And second, because at some point you have to account for the consequences of choices that were made, even if they could have been different. Thomas Markov writes "A creature has only been deprived of the ability to move when they cannot choose to move, or could not have chosen to move." Thus he contends that even if the mind whipped creature chose not to move, they have not been deprived of movement after the fact, because they still could have chosen differently. For an absurdly extreme version of this argument, consider a flying creature that chooses, on its turn, to do something that renders it unconscious through damage (drinks a potion of poison, casts life transference, whatever). The unconscious condition means that it "can't move". I don't accept the argument that it hasn't been deprived of movement simply because the action to render itself unconscious was its own choice, and that even while unconscious it maintains itself aloft because it could have chosen differently. Whether or not it falls should not depend on whether its unconsciousness was a result of its actions or someone else's.