Although movement in 5e is not governed by squares, they are probably a helpful model to this particular exercise. Everything in the universe is effectively measured in 5' increments and using squares to model this will give us an effective answer to this question Let's posit a 50' wall

 x
 x
 x
 x
 x
 x
 x
 x
 x
 x

Now let's look at the veritcal space next to the wall

 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx

Finally we need to have the flat part on top of the wall, and the space above that

  yyyyyyyyy
 yxxxxxxxxx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx

Now, supposing our hero starts at the first y (marked a below), and runs up, he runs out of movement at the 10th y (marked b below).

  yyyyyyyyy
 bxxxxxxxxx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 yx
 ax

If you start at a, you might be able to climb up onto the top of the wall. However, if you have a 45 foot wall you are safe. If you have to move to a, a dash would be required to not fall (as you don't have the ability to stay vertical).

The problem here is that if the wall extends at all into the square above the one marked b, then you're in trouble, you have to move into the square above it as diagonal movement is not possible.

This will be both at the discretion of the map makers (if the building extends a bit above the square you're probably not going to move diagonally) and also your DM (he may allow a bit of wiggle room here). Consult your DM before you attempt any 50' wall climbs.

45' wall climbs (and lower) are safe though.

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.