How to adjudicate this depends on what kind of trigger the monster uses.
When a creature has triggered actions in its action block, those actions are usually annotated with some type. You might see it as an immediate reaction, immediate interrupt, opportunity action, or free action. They also specify the triggering event or action, such as "another creature ends its move adjacent to you" or "another creature moves adjacent to you".
Yes, these are different triggers. The full breakdown goes as follows:
A charge is a single action with multiple "events".
Charge a Target
Action: standard action. [... F]ollow these steps.
1 - Move: The creature moves up to its speed toward the target. Each square of movement must bring the creature closer to the target, and the creature must end the move at least 2 squares away from its starting position.
2 - Attack: The creature either makes a melee basic attack against the target or uses bull rush against it.
Rules Compendium, p. 240
Any other power that says it can be used as a charge attack still follows these steps. The move ends before the attack happens, but for purposes of triggering enemy actions, the charge action is composed of several events.
Each square of movement during the charge is its own separate event.
Ending movement before you attack is an event.
Targeting the creature with an attack is an event.
Hitting or missing the attack is an event.
Dealing damage (if any) is an event.
Triggers all resolve differently.
From most to least restrictive... generally:
A creature can use an immediate reaction after the triggering action or event has completed. It can only use one immediate action before its turn comes up again.
A creature can use an immediate interrupt in response to an event, but the interrupt will resolve first; it can shield against an attack that would have hit it and the attack can then miss its boosted AC. It can only use one immediate action before its turn comes up again.
A creature can use an opportunity action with the same timing as an immediate interrupt, but it can take one opportunity action on each creature's turn.
A creature can use a free action with the same timing as an immediate interrupt. It can take any number of free actions on each creature's turn, but only one of them can be an attack.
Putting it all together...
A triggered action that reacts to "when another creature moves adjacent" is not reacting to the end of that movement, but rather the event of "moving adjacent" - one particular square of movement during the charge. The charge can continue if the charging creature has squares of movement left and can continue closing in.
A triggered action that reacts to "when another creature ends its move adjacent" is reacting to the event of the end of the move. The charge movement has already ended and cannot continue.
I can't claim to have access to all monster text, but I've looked through a few of my monster manuals. Monsters who have rules text about "ends its move" tend not to be using it in trigger text - they're skirmisher monsters who get a benefit from covering a lot of ground on their turn. Monsters that react to enemy movement tend to use "moves adjacent", like the level 1 Kobold Dragonshield.