Reactions come after their triggers, except when they don't
If the timing is specified in the ability, use that timing. Otherwise, reactions occur after the trigger.
This entire thing is a matter of specific over general. In 5e, when a specific rule and a general rule conflict, the specific rule wins.
Xanathar's Guide to Everything outlines the general rule for the timing of reactions:
If you’re unsure when a reaction occurs in relation to its trigger,
here’s the rule: the reaction happens after its trigger completes,
unless the description of the reaction explicitly says otherwise.
So, if the timing of the spell/ability you are using does not specify a timing then the reaction will come after whatever triggered it.
This is essentially just rephrasing what was already in the DMG since its release:
Various spells and features give a creature more reaction options, and
sometimes the timing of a reaction can be difficult to adjudicate. Use
this rule of thumb: follow whatever timing is specified in the
reaction’s description. For example, the opportunity attack and the
shield spell are clear about the fact that they can interrupt their
triggers. If a reaction has no timing specified, or the timing is
unclear, the reaction occurs after its trigger finishes, as in the
Ready action. (D&DB)
Thus, nothing has changed in the way reactions are handled since the DMG was released. Even with the rule being in the DMG and XGtE, you may (and probably will) still meet players who misunderstand how reaction timing works. There is not too much you can do to prevent this, but having a thorough understanding of the basics yourself will allow you to talk them through the issues with confidence that you are correct. Thus, I have prepared an extensive analysis below to hopefully help you accomplish this.
Dealing with Adventurers League and false expectations
Firstly, it is simply a fact that a DM is going to have to teach/correct that players that don't know or misunderstand the rules. It is one of the DM's primary jobs. There is not much you can do at the table to fix a character built on a misunderstanding of a rule whether it was based on not owning a particular reference material or not.
So how do you deal with it when it comes up? Well, as an AL DM, you are bound to follow the rules as written for all cases in which the rules are clear. However, if there was a case of a reaction ability that you found unclear lean on the general rule. If the ability is unclear then just assume it comes after the trigger. If assuming this seems to break the ability or make it useless then it is probably incorrect. In short, either go with the reaction happening after the trigger or go with what allows the ability to work.
Reaction spells
Below, I go through every currently published spell with a casting time of reaction and show how it either follows or is an exception to the general rule above. Remember, each one of these spells is analyzed using this same rule. As long as you understand it, there should be minimal issues with adjudicating most reaction timing abilities.
Shield - interrupts trigger
Trigger: when you are hit by an attack or targeted by the magic missile spell.
Until the start of your next turn, you have a +5 bonus to AC, including against the triggering attack, and you take no damage from magic missile.
So, shield explicitly says that it takes effect before the triggering attack resolves (and before the triggering magic missile spell hits). That means it is an exception and may take effect before the triggering effect resolves. This is an exception to the general rule because it's timing is specified by its effects.
The DMG also explicitly says that shield interrupts its trigger:
For example, the opportunity attack and the
shield spell are clear about the fact that they can interrupt their
triggers.
See this Q&A for more on the timing of shield: Do you have to cast Shield prior to your opponent's attack roll?
Trigger: when you take you take acid, cold, fire, lightning, or thunder damage.
The spell captures some of the incoming energy, lessening its effect on you and storing it for your next melee attack. You have resistance to the triggering damage type until the start of your next turn.
Again, the spell wording implies that it comes before the damage from the attack is resolved and thus it applies to the triggering attack as well. See this Q&A and this Crawford tweet for more.
Trigger: when you see a creature within 60 feet of you casting a spell.
You attempt to interrupt a creature in the process of casting a spell. If the creature is casting a spell of 3rd level or lower, its spell fails and has no effect.
Counterspell thus interrupts the spell being cast explicitly (also, if it did not it would literally be useless).
This post deals partially with the timing of counterspell: Can a readied spell be Counterspelled after it is cast, but before the trigger occurs?
Trigger: when you or a creature within 60 feet of you falls.
A falling creature's rate of descent slows to 60 feet per round until the spell ends.
The effect description says that it slows creatures that are in the process of falling. This means that it interrupts the falling and is thus another exception. Again this is also clear because if feather fall did not interrupt the falling creature's decent, it would be useless as the ground will have stopped the creature already with potentially deadly results.
Trigger: being damaged by a creature within 60 feet of you that you can see.
You point your finger, and the creature that damaged you is momentarily surrounded by hellish flames.
Hellish rebuke does specify timing, but it aligns with the normal rules of reactions. The trigger is being damaged and the reaction occurs after that damage is resolved. So, this spell is not an exception.
See here for further discussion: Hellish Rebuke reaction with zero hitpoints
Soul cage - occurs after trigger
Trigger: when a humanoid you can see within 60 feet of you dies.
This spell snatches the soul of a humanoid as it dies and traps it inside the tiny cage you use for the material component.
The trigger is a humanoid dying and the spell snatches the soul as soon as its death is complete. Thus, this reaction does not interrupt the trigger and happens after the trigger is resolved.
Non-spell reactions
I'll go over the two most commonly used non-spell reactions. There are many abilities and other effects that are reactions so it would be incredibly unwieldy to go through and analyze them all. However, knowing and understanding these two should cover a large portion of your reaction timing issues in most games (in my experience).
Trigger: variable
When the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.
The ready action can be difficult to adjudicate because it is up to the person readying the action what they want the trigger to be. What makes this especially tricky is that there are no real guidelines for how specific of a trigger you can set which leads to some confusion when it comes to, for example, the question of whether you can interrupt the casting of a spell with a readied action (also see this).
One thing is quite clear though, the readied action is always carried out after the triggering condition is met. No matter how clever the exact trigger is worded, the readied action can never precede the triggering condition.
Trigger: when a hostile creature that you can see moves out of your reach.
To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.
An opportunity is thus taken before an enemy's movement completes and thus interrupts the trigger. The enemy is free to continue moving after the resolution of the opportunity attack if they are able.
The DMG also explicitly says that the opportunity attack interrupts its trigger:
For example, the opportunity attack and the
shield spell are clear about the fact that they can interrupt their
triggers.
These are all the spells with a casting time of reaction that are currently published and a couple examples of non-spell reactions.
You don't have to look at this list though as long as you remember adjudicating every single one of these things can be done just by looking at the particular ability by following one simple rule: if a timing is listed or implied use that timing, and have the reaction resolve after the trigger for all the others.
