Say a warlock is already on an independent mount (i.e. mount chooses where to move) and then uses the One with Shadows eldritch invocation to turn invisible. On the mount’s initiative, it moves; does the warlock turn visible?
Alternately, if the mount is being controlled, does that change the outcome?
The problem here comes down to how "move" is interpreted in this phrase:
you can use your action to become invisible until you move, take an action or a reaction.
The problem here is that, in plain english, "you move" can be interpreted as saying "you actively move" or "you are moved". Luckily we do have a tiny bit of context that might point us to the intended meaning. When you "take an action or a reaction" you are doing so actively using your action or reaction. The fact that move is combined with these two could be read as evidence that "move" is also intended to be interpreted in that way.
We also have some indirect evidence from other abilities that show that if they had intended "move" to mean "changes position at all" they would have phrased it differently. For example the rule for Opportunity Attacks says:
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.
And the description for Dwarven Plate:
[I]f an effect moves you against your will [...]
In this case, the rule for OAs explicitly calls out that it applies to things that move your character without using your movement. For example, falling by gravity, being pushed by an explosion and movement on a mount would all be considered things that "move you" but not you "moving".
On the other hand, every instance of the term "moves" or "you move" seems to be talking about moving using your movement. Spells and other effects that move a creature without them using their movement generally use terms like "push" or "shove" and not "move".
There is enough unclarity here where a DM is going to make a ruling, but I think it is a reasonable RAW interpretation, all things considered, to say that "you move" here is referring to you using your movement to move and not other things changing your position.
Even if the warlock is invisible, the mount, by default is not. So under most circumstances I'm not even sure how this would help the warlock too much. They would retain the benefits of being invisible, but, on a visible mount, their location would be known.
Additionally, the restriction on actions and reactions are still in effect so at most they would be able to use one of those before the invisibility breaks. This doesn't really seem like anything too powerful such that a DM would need to rule against the player. But, depending on the game you are playing I think there is room for a DM to comfortably make either ruling and still technically being within the wording of the rules.
One serious issue with reading the ability this way is that the warlock could use the action to become invisible in an area of low light, then use the mount's movements to move out of the light and still technically be invisible. This, at least, seems clearly against the intent and flavor of the spell. If a DM wanted to follow the above ruling this would be something they need to take into consideration.
Jeremy Crawford seems to support this reading in this tweet saying:
"Enters" and "moves into" mean enters and moves into, respectively, in D&D. They have no special game meaning, other than that "moves" refers to movement. "Enter" is more open-ended.
However, this seems to contradict an earlier ruling about spike growth that seems to imply that "moves" does not require a creature to actually spend movement.
Spike Growth says:
when a creature moves into or within an area [...]
About which this question was asked and answered:
Q: If you shove/push a creature into Wall of Fire, Spirit Guardians, Spike Growth, etc, do they get affected?
Crawford: A push is an effective way to force a creature to enter an area of effect, unless it requires willing movement.
Crawford does not give an explicit yes or no to the question but he seems to imply that pushing a creature into spike growth would cause it to be affected thus further implying that "moves" can be made without the use of movement.
However, both these rulings are relatively vague and indirect indications of intent. And Crawford has been contradictory/inconsistent in many rulings over the years so it is even unclear if both of these statements are intended to be in agreement or not.
If your DM/table like to consider designer intent in your rulings, feel free to take this into consideration, regardless of how you interpret them.
No matter which ruling you are considering, the state of the mount does not matter. Whether controlled or independent, a mount does not use your movement to move you around and it still changes your location. Thus whether a mount is controlled or independent does not affect this ruling one way or another.
This is the description of One with Shadows
When you are in an area of dim light or darkness, you can use your action to become invisible until you move, take an action or a reaction.
If you move, no matter how you move, the invisibility ends.
