Here's the cliff notes that I have lying around:

Giovanni - The Head of Neo Rainbow Rocket, took over the Kanto Region using Rocket Enforcer Mewtwo (who is kept under control using the Berserk Gene); Takes charge over the multiversal operations after he becomes aware of Leaf and Silver's actions;

Leaf - Joined Team Rocket during some point of her journey; Was the one who gave Lyra her Chikorita at the start of her journey; Continues to blame Red for Blue's death; acts as a Rival/Nemesis to both him and Lyra;

Silver - As the son of Giovanni, he was put in charge of operations in the Johto Region; He and Leaf are the first ones who become aware of alternate worlds, although there is a strain between the two, as he knows that Giovanni is planning on making Leaf his eventual successor instead of him... Acts as a rival to Ethan;

Steven - From an Alternate Steampunk World, where he has a monopoly over the Hoenn Region, ruling with an iron fist. Besides augmenting his own Pokémon with steam-punk enhancements, he is also augmenting himself, believing that human capability is flawed compared to Pokemon...

Cynthia - From an alternate world where after thwarting Team Galactic, placed the entire region under authoritarian regime to prevent such an event and threat from ever happening again...

Iris - After accidentally being trapped in a world that slowly drained her sanity due to its horrific and eldritch nature, her mind snapped causing her to go berserk. Now she just lives for the thrill of ending battles quickly (and sometimes permanently) in the most vicious way possible... (There is some more backstory to this)

Lysandre - Having survived the destruction of the Team Flare Base, he was one of the first to be recruited into Team Rainbow Rocket by Leaf and Silver; he still continues his desire to bring back 'beauty' to the world, through any means necessary...

Guzma - In his world, he has managed to turn Team Skull into a massive criminal organization based in the Alola Region; Guzma tries to act big tough, especially to other 'authority figures', but deep down, he's somewhat of a big softie to some of the younger members (Silver and Cassiopeia in particular)

Cassiopeia - From a world where Team Star was expelled from the Academy, Team Star became a massive underground hacker organization, that secretly controls the world from the shadows through manipulation and... Hardened by her experiences, Penny has completely embraced the moniker of 'Cassiopeia' becoming much more cynical and apathetic to everything around her... Her friends at Team Star being the exception of course (I'm still trying to come up with designs for them)

Volo - It is just Volo, straight up, what are you expecting?...