From the Super Mario Wiki
A Beach Koopa is a Koopa Troopa that has lost its shell and wears nothing more than an undershirt (and/or in some cases, boxer shorts), first appearing in Super Mario World. In their original appearance, Beach Koopas appear small and hunched over, but were later depicted as being able to walk around like normal, shell-wearing Koopa Troopas. Other species of Koopas are also referred to as "Beach Koopas" when they lose their shells.
While technically not Beach Koopas, in the original arcade version of Mario Bros., if the Shellcreepers are left alone, not attacked, while they are upside down, they will pop out from their shells to flip back their shells. When this happens, they highly resemble Beach Koopas, wearing a white shirt and shorts.
Super Mario series
Super Mario World
When first knocked out of its shell, the Beach Koopa is extremely vulnerable. If given enough time, it will right itself and pace forwards in search of a new shell. They act very similarly to their shelled counterparts. Green ones fall off cliffs, Red ones and the slightly faster, Blue ones try to avoid falling, while the Yellow ones act like a quicker Green one, but will also follow Mario or Luigi. Green and Red Beach Koopas will become the color of the shell they enter. Yellow Beach Koopas will attempt to jump into any nearby shell; this also allows them to dodge shells thrown at them. Once a Yellow Beach Koopa enters a shell, it will become a Kamikaze Koopa: the shell will flash rainbow colours and spin around after Mario. Blue Beach Koopas are visually distinctive with their muscular legs and will kick any shell they come into contact with even if it is moving; they will also kick disabled Galoombas, Bob-ombs, flashing Grab Blocks, and Buzzy Beetle shells in their way. Blue ones are also capable of sliding against Mario. If two Beach Koopas enter the same shell simultaneously, they will become only one Koopa Troopa. Any Super Koopa who wields a flashing cape will revert to a Beach Koopa after being stomped on and relinquishing its Cape Feather. Its artwork appears as a stamp in Super Mario 3D World.
Super Mario 64/Super Mario 64 DS
In Super Mario 64 and Super Mario 64 DS, Mario or others can attack a Koopa Troopa to knock its shell away, transforming the Koopa into a Beach Koopa. If attacked again, the Beach Koopa would be destroyed and a Blue Coin would pop out. If not defeated, the Beach Koopa will frantically attempt to dive back into its shell even if Mario is riding it sometimes killing itself. If the naked koopa is seen again after the shell is destroyed he will act as if he had an imperceptible shell.
Super Mario Sunshine
In Super Mario Sunshine, Electro-Koopas become Beach Koopas for a short time when they throw their shell at Mario. If Mario manages to jump on the Koopa or squirt it with water, so it is electrified by its own shell, it will be defeated. These Koopas have pink boxer shorts.
Super Mario 3D World
After more than six years of absence, Beach Koopas reappear in Super Mario 3D World. Unlike other games, they share the same behavior as regular Koopa Troopas in the game, except they have an angry look. If the player keeps them away from their shells long enough, they'll stop chasing the player and will become sad.
Super Mario World television series
Only appearing in the Super Mario World episode "Fire Sale", the silent, Koopa Troopa henchmen of Kootie Pie Koopa is continually knocked out of his shell, becoming a Beach Koopa, in the episode, usually during one of Kootie Pie's violent temper-tantrums.
Mario is Missing!
Although normal Beach Koopas do not appear in Mario Is Missing!, at the end of the MS-DOS version, Luigi removes Bowser's shell. When he does so, Bowser is shown to be wearing polka-dotted boxer shorts, similar to a Beach Koopa.
Beach Koopas are also found in Super Mario World 2: Yoshi's Island, Yoshi's Island: Super Mario Advance 3 and Yoshi's Island DS, although here, they walk upright when evicted from their shells, instead of becoming small and hunched over, as in Super Mario World.
Super Princess Peach
Super Smash Bros. Brawl
In Super Smash Bros. Brawl, a Beach Koopa briefly appears in the Subspace Emissary cut scene when Donkey Kong smashes a Koopa Troopa's shell with his fist, leaving a Beach Koopa. However, it is near impossible to see since it is only shown for half a second and it is very dark.
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