This article contains fan speculation. There is no solid evidence for or against some parts of this article.
There are many NPCs in Pokémon that do not serve much of a role beyond progressing the plot or handing the player an item. As a battle enjoyer myself, I think that's a shame! Here are some battles I cooked up for a few of these oft-neglected Trainers, with quotes along with the point in the story you'd face them at!
For his rematch team in FireRed and LeafGreen, he includes the unobtainable Johto first partner Pokémon, as well as Murkrow, who parallels Agatha's Misdreavus, being two version-exclusive Johto Pokémon obtainable on the Sevii Islands. Murkrow is also only obtainable in Kanto in the Generation II games. The three Johto first partner Pokémon reference Oak's connection to Johto and to imply that there is a vast world waiting to be explored just beyond the Kanto region. Finally, his ace, Nidoking, references the opening cutscene, later confirmed to be Oak battling against Agatha.
His team in Let's Go, Pikachu! and Let's Go, Eevee! is similar to his FireRed and LeafGreen rematch team, though with Alolan Persian in place of Murkrow, and Kangaskhan in place of the Johto first partner Pokémon. Kangaskhan's inclusion is a reference to his Adventures counterpart.
"Good to see you, <player>! Congratulations on becoming the PokémonLeague champion!"
"How is your Pokédex coming along?"
"You know, back in my day, I used to be quite the battler!"
"I'd love to see just how strong you've grown firsthand."
"What do you say?"
No:"I see."
Yes:"Get ready! I taught my grandson everything he knows!"
Being defeated
Upon being defeated: "Incredible! You are a Pokémon master!"
After being defeated
"You know, there's more ways of being a great trainer than winning Pokémon battles."
"Your journey does not have to end here. Exploring to your heart's content, meeting new people and Pokémon... that's what being a Pokémon master is really about!"
"Thank you for the amazing battle! If you catch any more Pokémon, please allow me to see your Pokédex!"
"You've grown so much since you first left Pallet Town with that <first partner> of yours."
"It seems like there's no limit to how much you're able to grow. I want to see just how much stronger you've gotten!"
"Let me see the power of a PokémonLeague Champ one last time!"
No:"I'll be waiting until you're ready, then."
Yes:"Show me all that you've learned on your journey!"
Being defeated
Upon being defeated: "To think I was lucky enough to witness the beginning of your legend!"
After being defeated
"I'd given up on Pokémon battling to dedicate my time to creating the Pokédex."
"Back then, I couldn't imagine having enough time to pursue both of these dreams, but now I've seen a fantastic trainer who also has the makings of a fantastic PokémonProf.!"
"An old friend tried to teach me that once. She's a member of the Elite Four now. If you see her, please give her my regards!"
"Even for a PokémonProf. like me, there's no shortage of knowledge to be gained. You, too, will have many opportunities to learn."
"No matter what kind of trainer you want to become, don't ever lose sight of your dream!"
"You've grown so much since you first began your journey with PikachuP/EeveeE. I can tell you've become a lot stronger!"
"You wouldn't know it by looking at me, but I used to be quite the battler when I was your age!"
"I'd love to see just how strong you've grown firsthand."
"What do you say to a battle?"
No:"I'll be waiting whenever you're ready!"
Yes:"Get ready! I taught my grandson everything he knows!"
Being defeated
Upon being defeated: "Incredible! You're one of a kind!"
After being defeated
"The way you and your Pokémon battle... it reminds me of an old friend! But there's more ways of becoming a great trainer besides winning Pokémon battles, you know."
"Some trainers prefer to explore, meeting new people and Pokémon. Others seek out thrilling, heart-pounding battles. Myself, I prefer conducting research on Pokémon."
"No matter what path you choose to follow, you're destined to make your dreams come true!"
"I hope you continue to have fantastic adventures with PikachuP/EeveeE for many years to come!"