This one will be long...
Overlord
Ainz Ooal Gown
Rarity:
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Puppet Fighter:

Minions:

(White)
Rules:
Magic and PSI Unleashed
Items: N/A
Stage: Castle Siege (Throne Room)
Conditions:
The enemy's magic has increased power
The enemy will occasionally be invincible
You'll occasionally take sudden damage
Music Track:
Clattonia
Spirit Type: Attack Primary
Spirit Effect: Magic Attack ↑
Overlord
Albedo
Rarity:
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Puppet Fighter:

(White)
Minions: N/A
Rules:
Hard to Launch
Items: N/A
Stage: Castle Siege (Throne Room)
Conditions:
The enemy's magic has increased power
The enemy has increased defense
The enemy has super armor and is hard to launch or make flinch
Music Track:
VORACITY
Spirit Type: Defense Support
Spirit Effect: Easier Perfect Shield
Overlord
Shalltear Bloodfallen
Rarity:
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Puppet Fighter:

(Purple)
Minions: N/A
Rules: N/A
Items: N/A
Stage: Dracula's Castle (Omega form)
Conditions:
The enemy is metal
The enemy has increased attack power
The enemy reflects projectiles
Music Track:
L.L.L.
Spirit Type: Grab Support
Spirit Effect: Lifesteal
Overlord
Cocytus
Rarity:
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Puppet Fighter:

(Blue)
Minions: N/A
CPU Alley:

(Red)
Rules:
Hazard: Ice Floor
Items: N/A
Stage: Great Bay (Omega Form)
Conditions:
Join forces with a CPU ally
The floor is covered in ice
Music Track:
GO, CRY, GO!
Spirit Type: Defense Support
Spirit Effect: Ice-Floor Immunity
Overlord
Aura Bella Fiore
Rarity:
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Puppet Fighter:

(Pink)
Minions: N/A
Rules:
Item Tidal Wave
Items: Assist Trophies
(Chain Chomp, Klaptrap, Rathalos, Tiki)
Stage: Arena Ferox
Conditions:
Hostile Assist Trophies will appear
The enemy is easily distracted by items
Music Track:
The dark elf twins
Spirit Type: Grab Support
Spirit Effect: Lightweight
Overlord
Mare Bello Fiore
Rarity:
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Puppet Fighter:

(Yellow)
Minions: N/A
Rules:
Earthquake
Items: N/A
Stage: Spear Pillar (Omega form)
Conditions:
Periodic earthquakes will shake the stage
The enemy's FS Meter charges quickly
Music Track:
A sacrifice of 70 thousand
Spirit Type: Attack Primary
Spirit Effect: Magic Attack ↑
Overlord
Demiurge
Rarity:
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Puppet Fighter:

(Red)
Minions: N/A
Rules: N/A
Items: N/A
Stage: Unova Pokémon League
Conditions:
The enemy's air attacks have increased power
The enemy has super armor and is hard to launch or make flinch
Survive until the timer runs out
Music Track:
Oath of Loyalty to the Supreme Being
Spirit Type: Attack Support
Spirit Effect: Air Attack ↑
Overlord
Sebas Tian
Rarity:
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Puppet Fighter:

(Persona 4 Protagonist Wig + Butler Outfit, 2122)
Minions:

(Black)
Rules: N/A
Items: N/A
Stage: Town and City (Night)
Conditions:
The enemy's arm and leg attacks have increased power
Reinforcements will appear during the battle
Defeat the main fighter to win
Music Track:
Gray old gentleman
Spirit Type: Attack Primary
Spirit Effect: Fist Attack ↑
Overlord
Yuri Alpha
Rarity:
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Puppet Fighter:

(Black)
Minions: N/A
Rules: N/A
Items: N/A
Stage: Castle Siege (First form)
Conditions:
The enemy's physical attacks have increased power
The enemy favors grabs and throws
Music Track:
MMO RPG YGGDRASILL
Spirit Type: Grab Support
Spirit Effect: Mr. Saturn Equipped
Overlord
Lupusregina Beta
Rarity:
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Puppet Fighter:

(Red)
Minions: N/A
Rules: N/A
Items: N/A
Stage: Smashville
Conditions:
The enemy heals over time
The enemy has an extra midair jump
Music Track:
VORACITY
Spirit Type: Defense Support
Spirit Effect: KOs Heal Damage
Overlord
Narberal Gamma
Rarity:
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Puppet Fighter:

(Black)
Minions:

(Black)
Rules: N/A
Items: N/A
Stage: Lost World
Conditions:
The enemy's electric attacks have increased power
The enemy will charge up a powerful Final Smash
Music Track:
NPC Battle
Spirit Type: Attack Primary
Spirit Effect: Electric Attack ↑
Overlord
CZ2128 Delta
Rarity:
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Puppet Fighter:

(Lacy Headband + Maid Outfit, 2212)
Minions: N/A
Rules: N/A
Items: N/A
Stage: Reset Bomb Forest (Omega form)
Conditions:
The enemy's ranged weapons have increased power
Music Track:
Army of death
Spirit Type: Defense Support
Spirit Effect: Made of Metal
Overlord
Solution Epsilon
Rarity:
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Puppet Fighter:

(Yellow)
Minions: N/A
Rules:
Hazard: Sticky Floor
Items: N/A
Stage: Luigi's Mansion
Conditions:
The floor is sticky
Music Track:
NPC Battle
Spirit Type: Grab Support
Spirit Effect: Poison Damage Reduced
Overlord
Entoma Vasilissa Zeta
Rarity:
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Puppet Fighter:

(Purple)
Minions: N/A
Rules: N/A
Items: N/A
Stage: Unova Pokémon League
Conditions:
The enemy can unleash powerful critical hits at random
The enemy has increased attack power when the enemy's at high damage
Music Track:
NPC Battle
Spirit Type: Grab Support
Spirit Effect: Undamaged Attack ↑
Overlord
Pandora's Actor
Rarity:
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Puppet Fighter:

(Yellow)
Minions: N/A
Rules: Item Tidal Wave
Items: Equippable Items
Stage: Kalos Pokémon League (Dragon chamber)
Conditions:
The enemy starts the battle with a Poké Ball (Ditto)
The enemy is easily distracted by items
The enemy loves to taunt
Music Track:
Wenn es meines Gottes Wille
Spirit Type: Defense Primary
Spirit Effect: Transformation Duration ↑
Overlord
Hamsuke
Rarity:
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Puppet Fighter: Giant

(Libre)
Minions: N/A
Rules:
Hazard: Strong Wind
Items: N/A
Stage: Mushroom Kingdom U (No hazards)
Conditions:
Dangerously high winds are in effect
The enemy's tail attacks have increased power
Music Track:
A day in Nazarick
Spirit Type: Grab Support
Spirit Effect: Strong-Wind Resist