Dark Moon

Abridged move description:

This is a so-called bombardment kick move, with the kick delivered while flipping forward in mid-air. This is one of the more unusual moves in the Hazuki style, considered heretical in ancient Jujitsu... The reason this type of move was developed is found in the training system of the Hazuki Style. Among the throwing moves to be mentioned later is one called the Darkside Hazuki which involves spinning the opponent 180 degrees and dropping him head first. ...the person on whom the move is being practiced adopts a defense of spinning in mid-air before landing. That move, with a kick added, is the Dark Moon.

Because the practitioner himself falls after executing the move it is easy to leave an opening for the opponent, but those who have mastered the move can quickly mount a defense to solve this problem.

This move does exist in real life and is known Do Mawashi Kaiten Geri in Japanese or
as the rolling thunder or Kyokushin wheel kick. This move is typically seen in Kyokushin Karate and kickboxing competitions (typically thrown by kickboxers with a Kyokushin background). It is a very powerful kick as the user is throwing their entire body into the move and usually stuns or knocks out the opponent even it connects.

Peter Graham knocking out Badr Hari with the rolling thunder in the last minute of a fight he was clearly losing.

In Kyokushin Karate and kickboxing competitions, there is little risk as strikes to a grounded opponent are illegal. This can even be gamed for an advantage. Young kickboxing phenom Tenshin Nasukawa rarely lands the rolling thunder in his professional kickboxing matches (most of his knockdowns with them have been against outclassed MMA fighters). However, as the referee halts the action to let Tenshin get back up after flopping on the ground from a missed kick, he can run out the clock and buy a few seconds as a breather before he stands back up

Tenshin knocking down the overmatched Yusaku Nakamura during a Rizin match

In MMA, missed or botched connections are much riskier as you are grounded while your opponent can easily score a free strike or can engage in grappling and secure top position. There are also few Kyokushin strikers who transition to MMA in comparison to the amount that end up competing in kickboxing, which may also explain the move's rareness in MMA competition.

Though it does still happen on occasion in MMA as former UFC veteran Ross Pearson found out the hard way

The description mentions the move being considered heretical in jiujitsu. Traditional Japanese jiujitsu is designed with the use of weapons and armored opponents in mind. In modern grappling competitions, not all positions on your back are considered bad--many submission grapplers even prefer to engage from their back rather than top position. However, in traditional Japanese Jiujitsu, being on the ground on your back is terrible as you're one step away from getting a knife shoved through your eye. It's logical that the move would be considered heretical in that context as you're purposely tossing yourself to the ground for little reward as the other guy is probably wearing armor, making most unarmed strikes ineffectual.

This picture courtesy of historical German fencing master Talhoffer depicts the concept of why being on the bottom = bad when weapons are involved.

I'm skeptical of the in-game logic of the Dark Moon originating as a by-product of safe practicing of the Darkside Hazuki move. If the kick was developed from the motion of the uke when thrown, then the Dark Moon should resemble the forward roll version of the rolling thunder rather the side roll version that Ryo uses (going off the in-game depiction of the Darkside Hazuki). Of course this could just be a simple continuity error between the description and the animation and not a lore problem or whatever. The connection between "roll-through to not get spiked on the head" to "hey what if I kick somebody really hard with this" is a bit dubious as well, but hey it wouldn't be the first move to be discovered via accident or messing around.

Darkside Hazuki, footage by Damp Macaroni

Forward roll variation of the rolling thunder demonstrated by Harold Howard during an early UFC match.

Big thanks to Suka Pass for allowing me to make a video of the move and get the description without having to download an emulator or buy a screen recorder.