As a refresher, RYUSEITAI is one of Yumenosaki's oldest units. Chiaki's RYUSEITAI was formed in Meteor Impact when the delinquent RYUSEITAI members ditch Chiaki and Kanata after the Battle of the Sea God. AKATSUKI was formed after the events of Crossroad.

Souma's recruitment

Souma and Keito have known each other even before Crossroad: Souma's a trouble maker. Still, Keito sees his potential.

KEITO
If he were properly produced, he’d become an interesting idol, wouldn’t he?
Crossroad (citrinesea)

Souma is there to witness Keito perform at DEADMANZ Live.

SOUMA
Also… Hasumi-dono, who I deeply respect, looks to be standing on stage. I have never seen that personage sing before, so I’m quite interested.
His voice is clear, and yet solid and beautiful. Ah, it feels good to the ear.
Crossroad (citrinesea)

Kuro and Keito decide to form a unit together. When Souma catches wind of it, he pursues it relentlessly.

KEITO
In any case, Kanzaki, I have told you many times that Akatsuki is nothing like you imagine [...] You should thrive in another unit that suits your strengths: join one or make another.
SOUMA
No. A sword may not choose its wielder, but I want to become an idol under you, Hasumi-dono [...] Now come! Give me the entry test again!
Shinsengumi (kotofucious)

At Kuro's suggestion, Keito gives in and allows Souma into AKATSUKI.

KEITO
I understand your point, Kiryu. I see there is no other choice. It's true we are in need of help, and I do want to develop Kanzaki's talent.
[...] I accept your admission, Kanzaki. Welcome to AKATSUKI.
Shinsengumi (kotofucious)

Midori's recruitment

Midori accidentally entered the idol course.

MIDORI
[...] It was only by mistake that I took the idol course in the first place... Then I somehow managed to pass it, and in the end I had no choice but to go to this academy.
Climax (310mc)

Given his attitude, Midori's recruitment might seem surprisingly easy: he simply went with the flow.

MIDORI
As soon as I entered the school, Morisawa-senpai came to my classroom and said stuff like “Ossu! I’m Morisawa Chiaki, the captain of the Basketball club! And, starting today, you’re a member of said club! Nice to meet you!”
He forced me to join, and turning him down would be a pain and all…
And I thought maybe he’d stop chasing me to my classroom if I joined the Basketball club, so I felt relieved for a second…
But then he was like “Takamine, won’t you become a hero? You’ve got the qualities of a hero! There is no mistake, you’re meant to be!”
So I ended up joining Ryuseitai too…
Sparkly First Years (meatball124)

Shinobu's recruitment

Shinobu was recruited around the same time as Midori.

SHINOBU
(Taichou-dono invited me, asking if I wanted to become a hero.)
(Well, at first I got scared of how close Taichou-dono was to me and thoughtlessly ran away.)
(But he appeared wherever I went and invited me each time, so I ended up nodding and accepting his offer.)
(So, surprisingly enough, I quickly decided which unit I wanted to join.)
Sparkly First Years (meatball124)

Tetora's recruitment

As he did with Midori and Shinobu, Chiaki had his eyes on Tetora early on.

MIDORI
No, I don’t know… Huh…? Then, another unhappy first year is going to become your victim…?
Being a member of this stupid group is super pitiful…
CHIAKI
Don’t say that. There’s a kid called Nagumo Tetora in your class, right? I invited him, and now I’m waiting for him to respond.
My hero sensors detected him! If that kid joins us, Ryuseitai will officially have five members! I’m really happy, mmhmhmm ♪
Sparkly First Years (meatball124)

However, Tetora auditioned for AKATSUKI first. After his rejection, Kuro guided him to Chiaki.

KURO
A man among men is someone like Morisawa. Someone who draws close the weak, protects them, and loves them. Become like him, like a hero of justice who appears in stories... A cool guy.
TETORA
Is that why you introduced me to RYUSEITAI?
KURO
I don't mind if you think so. Morisawa instead of me: RYUSEITAI instead of AKATSUKI.
Four Beasts of Fistfighting (healingbonds)

CHIAKI
“Remember how I said there was a kid I’d invited to Ryuseitai? Nagumo-kun has officially contacted me, saying he would join! Thus, we’re finally five people!”
MIDORI
“Ah, poor thing… Nagumo-kun couldn’t escape Morisawa-senpai’s pushy invitations either… My condolences…”
Sparkly First Years (meatball124)

REFERENCES
(1) Crossroad, !-Era, (Full story translations): acarnivalofstars, reikeiP
(2) Climax, !-Era, 310mc
(3) Four Beasts of Fist-Fighting, !-Era, Enstars Wiki, healingbonds
(4) Shinsengumi, !-Era, kotofucious
(5) Sparkly First Years, !-Era, meatball124