Once you are done in Aurora click on Ignis and hit battle. Bring along Oichi too, since Oichi is required in all your invasions against other nations. Once you enter, it's time to fight. The two guys you fought in Aurora are back, but this time, their leader is fighting along them. So it's 3 against 2, not so bad.
Lets analyze our enemies. Bidoof, The only non-fire pokemon in the battle, may be weak, but don't underestimate him, his headbutt not only does damage, but has a chance of flinching you, which will make the pokemon lose it's turn. Then we got Tepig and Chimchar, who both have Ember. This is a ranged attack that hits two blocks in front of them, plus Ember has a chance to burn your pokemon, very bad. so you're dealing with pokemon than can deliver nasty status effects.
Now let us look at the stage. As you can see, lava is about everywhere on the field. And only fire pokemon can go across it. Most of your enemies are fire types, so they can walk in lava as they please. Each stage does something to either help you or hurt you in the battle. In here, fire streams will pop out and in every few turns or less and block your path. Plus every few turns a volcano erupts and if your pokemon gets hit by a rock, it'll take damage, but this is luck based, so hopefully you won't get hurt in the process.
One more thing that you should notice are the hot springs. If you get in them, not only will you heal your pokemon, but if they were burned, the hot springs will cure it. Sounds good right? Do remember that the enemies can also use it to heal their pokemon if they were weak. And these fire pokemon can cross lava rivers to get to one, so this battle could take a long time.
Now that you got everything you need to fight these enemies. Go ahead and do so. Once you beat them, they will run away because you took over their kingdom. This will apply to warlords and warriors alike if you have met certain conditions. Although in the story, they'll run away, but few warlords and some warriors in this walkthrough are recruitable later. Then Kenji comes. He then explains how to get allies to join you. In the cave lies a Charmander and a trainer. You can tell when a pokemon is either wild or has a trainer with him before you you check the dungeon and see a face beside the pokemon. Defeat his Charmander in less than four turns and he will join you. Congrats, you got your first recruited warrior.
Kenji will congratulate you and go off on his own way. After that, two new nations will open up, Greenleaf, which is a grass type nation (hence its name) and Fontaine, a water nation. Feel free to take on either nation, but if you were smart, you would attack Greenleaf first because of your new fire pokemon. Or if you're feeling lucky, you can attack Fontaine and get that outta the way. Your choice.
Head to Greenleaf and burn down their grass pokemons, or if you're feeling very lucky, head over to Fortaine and show them who's boss.