Action Resolution: Starting and finishing an action in FFXIV comes in several distinct parts:

Step 1 depends on whether the action is instant cast or has a casting time:

Instant:
1a) During the first ~.5seconds of the animation (on your screen), checks to see if the user has kept facing with the target (if applicable). A snapshot is drawn from the moment the action queues and draws a frontal cone of about 45 degrees left or right of this direction. If the target moves out of this cone in the time between this and the next step, the action is considered to not have been started. To anyone else, it's as if you sat there doing nothing.
1b) If facing was kept (if applicable), the action queues successfully. HP/MP/TP or other costs are applied. Proceed to step 2.

Casting time:
1a) During the first ~10% (or ~.5seconds), checks to see if the user was moving. If yes, action fails. If no, server recognizes a cast was started and "tells" everyone. If facing with the target is broken before the "tell everyone" check, the spell also fails.
1b) The "middle" ~80%, where moving incurs the above "failure" condition but lacking proper facing does not.
1c) The last ~10% or ~.5 seconds. At this point the spell has essentially finished casting. You can move slightly at this point and the spell cast time still complete. Lack of proper facing towards the target at this point will cause spell failure. At this step, a snapshot of your current stats/buffs/debuffs and the target or targets' are taken for damage "calculation". HP/MP/TP or other costs are applied. Proceed to step 2.

2) Action animation is performed.
3) Damage numbers and/or buffs/debuffs are displayed. The time between the previous step and this varies by spell.
4) The effects of step 3 are "recognized" by the server, including movement of HP bars, application of said buffs/debuffs, and changes in enmity.

As an example, let's take the Black Mage's Fire 1.

1a) Fire starts casting
1c) Fire successfully casts
2) The "bolt" of fire is shot from the user, travelling to the target in about .5 seconds.
3) Fire deals (arbitrary) 700 damage, generates a stack of Astral Fire, and (arbitrarily) Firestarter procs.
4) The damage dealt is reflected on the target's HP bar, the Astral Fire stack and Firestarter buff are generated, and enmity from the attack is added to the target's enmity table.

The player can start acting again as early as when step 1c finishes. This means that by the time your first Fire hits the enemy, you're already partway through the next spell. And the delay between steps 2 3 and 4 is ultimately pretty arbitrary when viewed in a vacuum. There isn't a huge difference in any of it.

However, if the Black Mage were to die before step 4 happens (that is, after Fire's cast time succeeds, even after the damage numbers pop up, but before the change is "applied"), the entire thing will have been cancelled. Most Limit Breaks have a long animation time after their cast time, and are prominent examples of this behavior. Let's look at the Caster LB3: Meteor

1) The caster "paints" the target zone. Targets have NOT been decided yet
1a) Casting begins
1c) Casting is successful. AT THIS TIME the targets have been decided. They can be moved out of the target circle at this point and will still be hit. Conversely, targets that enter the area AFTER this point will NOT be hit.
2) The animation for Meteor begins, taking ~5 seconds.
3) Damage is dealt
4) Damage dealt is registered by the server.

In this case, there's a large gap in time between step 2 and the steps after. If the user dies in that time, the animation will be played but none of the effects will resolve because the user died.

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Let's take an extreme example: The Navel (Extreme)'s Titan. Heart phase comes and goes, but the slow Dragoon gets clipped by a bomb before Titan jumps away. He's now sitting at about 1600/4000 HP. Let's say the upcoming Earthen Fury will do about 2500 damage to him.

Now, the moment Titan reappears and Uses Earthen Fury, the damage for the action has already been decided. Yet, if that were the case in a vacuum the DRG would always die in this scenario. This is where effect resolution and animation time matter a bit. Earthen Fury takes about 3 seconds to resolve from the moment of his animation starting. Let's assume the healers were asleep until this exact moment for the sake of example.

1a) Earthen Fury animation begins
1b) Scholar uses Lustrate on Dragoon (will restore 1000 HP)
2) The animation for Lustrate is fairly short, completing and resolving in about a second. It resolved, Dragoon now has 2600 HP.
3) Earthen Fury resolves. Dragoon takes 2500 damage but is alive!

(to be continued)