After replaying Iconoclasts for a bit (not quite all the way through, mainly just parts of it) I noticed some things about the boss battles in which party members help out.
So, I figured, why not just make a list of all the things party members will do during boss battles (that I have noticed)? Which means that some of these notes will be obvious; but others, perhaps, not so much.
Not counting the battles in which you outright play as someone other than Robin, or the two tag-team battles with Mina!
-Kerthunk: (Mina)
For the first stage, Mina will follow you around and shoot at its weak spot when you reveal it.
In the second stage, she will shoot down at its weak spot regularly (which can, in fact, get it to the next stage without you doing anything), and duck shortly before it shoves the platform up towards the thorns, which, along with the boss’s own cue, allows you to duck in time yourself.
In the third stage, she will shoot at the controllers it spews out and destroy them, and hint at what you need to do in order to damage it.
-Kibuka: (Royal)
During the first stage, Royal will tell you what you need to do in order to a) help him reveal its weak spot and b) damage it; as well as causing an explosion that will knock it over if it’s close enough.
During the second stage, he will raise rocks with his powers and throw them at the boss, which (I believe) does more damage the less health it has, and does enough damage that he will actually finish it off on his own if you do nothing at that point.
Agent White: (Mina)
After the chase, when the battle begins, Mina will shoot at White, dealing a small amount of damage. As soon as he’s been damaged enough that he attempts to heal, she will punch a seed into him; after the first time she does this, his health bar changes colours, after the third time, he dies.
-Carver: (Elro)
Elro isn’t there during the first stage; but in the cutscene beforehand he shows you how to reveal its weak spot in order to damage it.
For the second stage, he will throw his sword at its now detached head, stopping it from spinning and allowing you to push it back. When it knocks his sword away, he will pick it up; after the first three times, it knocks his sword into the ceiling rather than the ground, and he won’t be able to do anything until you knock it down for him.
Now, I might be wrong, but I think I experienced a bug here; if you knock the sword down at just the wrong moment, Elro will throw it at the head - and then it vanishes, leaving the head free to continue attacking you with no way to defeat it. (This happened only once, and I’m bad at noticing things when I’m first fighting a boss, so it’s possible the sword somehow ended up on the ceiling again and I didn’t notice it before dying, but I don’t think that was the case.)
-Silver Watchman, third fight: (Mina, Elro)
While neither of them is able to attack the boss during this battle, they do move to avoid his attacks, which allows you to dodge them as well.
Omega Controller: (Royal)
Royal is unable to harm the controller and has no idea what to do; the person who hints at how to hold it back is actually the poor Concern soldier, who throws bombs at it before slipping off the platform.
Mother: (Mina, Royal, Elro)
Neither Elro nor Royal is in any condition to help out at the start of the fight. Mina will follow you, and shoot at the cat Mother is controlling, even though that does no damage. She will also shoot down the rocks Mother throws at you.
Once Mother is down, Mina will hint at what you need to do in order to bring her down so she can be damaged. Once she’s on the ground, Mina will run to her, and punch a seed into her. When Mother gets back up, Royal will attempt to throw a large boulder at her. As Mother’s telekinetic powers are stronger, it ends up flying in your direction instead, unless you destroy it. (Mina won’t attack it.)
For the rest of the fight, Mina will shoot at the rocks that Mother raises, but will stop trying to hit the cat. Royal will occasionally raise boulders to try and throw them at Mother (once per seed-punch, basically). It never works. (She overpowers him every time.) He will follow you as well, behind Mina. Whenever the cat opens its mouth to shoot at you, revealing its weak spot, both Mina and Royal will duck to avoid the attack.
(I’m very bad at the Mother fight, so most of this I got from watching videos of the fight on youtube, rather than observations while fighting her myself.)
Agent Black, chase and last battle: (Royal)
During the chase, while you are on the bottom floor, Royal will attack her with his magical explosion power; when she does the electric floor charge, he jumps to avoid it, and whenever she approaches him, he backs off. He will not follow you up to the other floors (which, from a mechanical standpoint makes sense, but story-wise, ???). Once the elevator is reactivated, he’ll run for the exit.
When the battle itself begins, he attempts to attack her, most likely whenever she stops moving for long enough. (If she moves once he’s started trying to attack, he will keep trying until he’s able to hit her.) He will back away from her whenever she comes near him, unless he’s trying to attack her; he will move closer if she gets further away. He still dodges the electric charges.
Once you get to the part where she‘s shuffling around, barely able to move, he stays put, for the most part; he will still back away and attack her if she shuffles in his direction. At this point, however, he actually does not attack unless she gets too close.
As for the Ivory Beast, he is unable to damage it, and is knocked out by one of its vine projectiles. He comes to during the final stage, and might be the one moving the elevator, although it is also possible that the Ivory Beast’s larger form was simply preventing the elevator from rising.
I read that if you do things right, Mina will join you for the final boss; I haven’t gotten that myself, however, so I have no idea what happens if she is there.
Royal, Searching For Robin in the Dark Cave: Spooky, blue eye-infested cave... Silver Watchman on the hunt. Still... No proof that Robin's in any real danger.
Robin, Being Chased By The Silver Watchman: Let me go!
