I was typing a long 4* farming primer for the new dungeons and closed the tab by accident.
Anyway, briefer version. This is merely a primer and not a proper guide as there's much more consolidated info out there (e.g. muketsu) I think cpp could chime in on droprates here. I'll be using the
same terminology cpp does in this post.
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What you should know with the new dungeon updates:
Elite bosses remains the best way to farm 4* orbs. Ideally you want to farm:
(1) A boss Battle that's also the first Battle of a Dungeon (best - e.g. Humbaba)
(2) *A boss Battle that is NOT the final Battle of a Dungeon (second best, because you can loop indefinitely* and not have to waste stamina on the previous stages each time - e.g. Asura in FFIV Feymarch)
(3) A boss Battle that is the final boss of a Dungeon, but the Dungeon uses little stam overall, and/or the boss Battle itself costs low stamina for airplane mode. (e.g. Phantom Train FFVI)
* Despite being able to loop the (2) case, you still want to use the airplane mode trick and only accept results if the 4* orb drops. That way you get refunded your abilities for attempts that don't drop the 4* which increases longevity, especially for difficult loops like Palumpolum.
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Dark 4* drops off the 1st Battle of Prima Vista FFIX, so it's a much shorter yet loopable 17stam track than the other Dungeons that drop Dark. Boss has 35k HP.
*** Phantom Train is 24 + 5(loop) vs Prima Vista's 17(loop) and is thus actually might be superior depending on the drop rates, even if it requires autobattling through the first 6 Battles. Whether it's worth it probably depends on the exact drop rates of each, and how much stam is worth to you at the convenience of having to not go through all the autobattling in Phantom Train. We'll have a clearer verdict with cpp's data.
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Earth 4* drops off 1st Battle of Nalbina FFXII, likewise for 17 stam.
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Lightning 4* drops off 2nd Battle (penultimate boss) of Palumpolum 2 FFXIII, for 18 stam. However, it's tricky as it's a 89 difficulty Elite dungeon which isn't the first stage like the previous two, so it's a little hard to loop indefinitely. 100k HP and is weak to Thunder (ironically the exact shit you want to farm). If you have Syldra, Ramuh, Thundaga and/or Thundaja, I think a legitimate course of action would be to go get the BM/Summoner Record Materia first which should up offensive staying power a shit ton. I'm -pretty- sure the Record Materia 2 refreshes 1 ability on both slots each, but don't quote me on that.
The boss is not immune to anything outside Death/Gravity though, so Intimidate will work wonders, even if you can't recharge it like you can the Thunder abilities. (Hilariously enough Intimidate ALSO uses 3* Lightning Orbs so happy lightning farming)
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Wind 4* Humbaba obviously
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Power 4* currently drops off Asura in FFIV Feymarch at the moment (Penultimate boss), but it's a PITA to get to the boss, and a far better option (future FFII, first boss) comes around later, so don't farm this now if you don't absolutely need to.
The rest don't have anything notable and/or soon. However Lightning/Earth/Dark are currently the most useful, so having all three in a far more farmable state come this dungeon update is good info to have.
Now I only need to master Elite Vector and plan minimum for Locke's Event will be met. Problem is I cannot even win the boss fight (so far). Gotta think of a better strategy. Then maybe I can try Thamasa... cause I will never even attemp Airship for what it is worth 