The monk must end its movement for each turn on solid ground or fall.
Unarmored Movement, PHB p. 78, emphasis mine:
At 9th level, you gain the ability to move along vertical surfaces and across liquids on your turn without falling during the move.
The monk doesn't fall during the move when moving this way. A character's move doesn't span turns; it's something they do during their turn. Thus, at the end of the monk's turn, their move ends, and if they aren't standing on something solid, they fall.
Also note the "on your turn" condition. A monk can't run over liquids or up walls when moving as part of a reaction during another creature's turn.
To address the examples in the question:
As an example, let's assume a monk with 30' movement is 10' away from a 30' wide body of water. The monk moves 10' on land and then 20' on water and then the monk's turn is over. Does the monk fall in the water?
Yes. Their move is over, so their Unarmored Movement ability isn't preventing them from falling. Also note that a 9th level human monk will have a speed of 45 feet.
A slightly different example. Let's assume the monk has 30' of movement and is at the edge of a 1000' wide body of water. How far can the monk move before falling in?
90 feet. The monk can spend one ki point and use Step of the Wind to take Dash as a bonus action, then take Dash again as their action for the turn, so they may move 90 feet in total that turn. Again, note that typically a 9th level monk would be able to move (3 x 45) = 135 feet.
One answer might be the monk can move its movement, then it is no longer able to stay above the water and gets dunked. If this is the case, can the monk get back up on the water and keep moving?
That's up to the DM. They might require some or all of:
- a Strength (Athletics) check to tread water as soon as their move ends
- a Strength (Athletics) or Dexterity (Acrobatics) check to get enough of their body out of the water to resume moving on top of the water on their next turn.
- using movement equal to half their speed to "stand up" from their "prone" position in the water.
To summarize: how far can a 9th level monk move along a vertical surfaces and across liquids without falling?
As far as they can move during their turn.
How the rules handle continuous movement
The rules actually don't specify how to handle continuous movement that extends beyond a character's turn. Very long falls should take more than a turn to complete, and a character who starts a jump at the end of their turn's movement presumably doesn't lose all momentum and stop moving.
So it's up to the DM; for jumping movement, they might rule that
- characters can only jump if they have enough movement left on their turn to finish the jump, or
- characters who end a turn in 'mid-jump' must finish the jump at the start of their next turn, and (maybe) can't take reactions until then.
Likewise for monks, the DM may rule that they can 'carry over' unused 'wall running' movement from turn to turn.
Of course, because of other aspects of the 5th Edition turn system, which allows characters to act, move, and be moved in between their turns, these options can introduce many other areas where the DM would need to make rulings. If a monk ends their turn walking on water, and they are moved by an effect such as thunderwave, are they still walking on water? Does the amount they were moved deduct from the distance they can move across water on their next turn? And so on.
My reading of the intent of the monk's Unarmed Movement feature is that it limits the distance a monk can travel up a vertical surface (or over liquid) without an intervening solid, horizontal surface. The easiest way to do that is to confine these intervals to the movement on the monk's turn.
The GameMastery Guide on Water says, "The rules presented in the Pathfinder RPG Core Rulebook for underwater combat apply to creatures not native to this dangerous environment [i.e. underwater], such as most PCs" (214). (Fortunately, monsters can be sorted by terrain.)
It seems reasonable that an eidolon that possesses gills be considered native to water, and, therefore, such an eidolon shouldn't suffer penalties when using its natural weapons during underwater combat, but you should check with your GM to be sure.
Also Paizo creative director James Jacobs in a 2012 post said, "Creatures that have the aquatic [sub]type don't have to worry about those [underwater combat] penalties[,] but all manufactured weapons must abide by these rules, regardless of whether or not the creature wielding them is aquatic." So, in conjunction with the GameMastery Guide's rule, if, for example, the eidolon opts for a greatsword instead its claws, the eidolon suffers the underwater combat penalties on attacks made with that weapon. (Note that Jacobs makes it clear that this is a ruling not a rule, but it seems okay and comes from an often authoritative source.)
Best Answer
The Aquatic Adventures campaign setting addresses several new rules for water environments, shows tips to survive underwater and how things works in different situations that is not covered by the core rulebook.
Freedom of Movement
Here are some new rules and clarifications for Freedom of Movement in the book:
So, on top of avoiding the weapon penalties for being underwater, you can also ignore some of the underwater mechanics, like swim checks to remain in place and not be pushed away by the water currents.
Pressure
Most aquatic creatures are adapted to survive a certain pressure threshold, but non-aquatic creatures do not have such defenses, and thus, they take pressure damage. This is reflected as 1d6 damage per minute under 100 feet underwater, and an additional 1d6 for every 100 feet after that. The DC to avoid this damage is 15 plus 1 per minute after the first. Depending on the native range of each creature, sudden changes could also result in pressure damage.
Freedom of Movement avoids this damage completely.
Currents and running water
Water sometimes is not static, its moving and characters swimming are being moved by water currents. Oceans and rivers work similarly in this matter, meaning that characters have to always make a Swim check with a DC based on the current strength, as described on the Swim skill. With a caveat, oceans are always at least rough water (DC 15), and faster ocean currents count at least as stormy water (DC 20).
Freedom of Movement removes the necessity of this check.
Off-balance and prone underwater
Yes, the core rulebook says that neither flying nor swimming creatures can drop prone. However, aquatic and swimming creatures can go off-balance when underwater, which works similarly to the prone condition: opponents gain +2 to all attacks against it and all attacks it makes take a -2 penalty using piercing weapons and cause half damage.
But how a creature becomes off-balance? By failing Swim checks. A trip attempt against a creature underwater forces them to make a Swim check with a DC equal to the result of the maneuver, and if they fail, they go off-balance. Many aquatic creatures can't be tripped, and can take 10 on swim checks, so this is hardly a problem to most of them. They can return their balance by making another swim check on their turn.
Freedom of Movement allows you to automatically keep your balance.
Buoyancy
This is the last, I promise. Basically buyoancy makes a creature go up or down when swimming. Whenever they fail a DC check by 5 or more, or when they do not spend any actions to swim, the creature either rises or sinks due to buoyancy. It has three normal states: sinking, rising and neutral.
Land creatures usually have a sinking buoyancy, unless they are carrying a substantial amount of low-density gear (like many wooden items) and little or no high-density gear (items made of metal or stone). The book presents air tanks and gear that can change your buoyancy on demand.
Buoyancy also has two extreme states: swiftly sinking and swiftly raising. If they are swiftly sinking, their carried gear is so dense that they can walk at the bottom of the body of water, and must make a DC 20 check to stay off the bottom. If they are swiftly raising, they must make a DC 20 Swim check to stay submerged, or they will move to the surface of the body of water.
The speed of this buoyancy-based movement depends on your Swim checks. Whenever you fail (or do not make) a Swim check, you move up or down 10 feet, and every turn after the first that you do not attempt to fix that, the speed increases by another 10 feet to a maximum of 30 feet per round after 3 rounds. When swiftly sinking or swiftly rising, a failure means you immediately move 30 feet in that direction. Also, while moving like this, the creature is considered off-balance.
When moving against your buoyancy, your movement is hindered, much like a flying creature flying upwards. As such, each 5 feet of movement in the opposite direction of your buoyancy costs you 10 feet of your movement.
Freedom of Movement allows you to automatically keep your buoyancy controlled.