When you say lost I don’t know if you mean actually die or just lost in the commotion, so I’ll do both. But these are kind of short, sorry
•If he just lost his s/o in the commotion, Kiba would panic. Of course he knows that battle is hectic and it’s not likely that he’ll be able to constantly keep an eye on them, or that they’ll be at his side the entire time. But he likes fighting next to his s/o? He feels like he has to protect them in any way that he can. So if he lost sight of them, it would serve as a distraction. He wouldn’t be able to fully concentrate or calm down until after the battle when he found them again.
•As for the s/o actually dying, Kiba probably won’t be able to carry on in the battle. At least not with a clear head. It takes someone encouraging him and forcing him to focus for him to turn his attention back to the battle. But he gets angry. He wants revenge. He can’t calm down. He’s devastated and furious and feels like he needs to kill something. It probably leads to him blindly running into battle and receiving a harsh injury in consequence
•Kiba blames himself for their death, and he probably doesn’t recover from the loss. It takes years for him to let it go, or at least, accept what happened. He tries to remind himself that his s/o died as a brave shinobi, and that he shouldn’t let his grief overwhelm him, because it’s an insult to their courage in battle. But he holds onto his anger for a long time too. Kiba’s hatred toward the enemy probably won’t go away until someone sits him down and tells him he can’t let it consume him. It’s not healthy and it’s not what his s/o would have wanted
•Kankuro is more likely to get mad at his s/o for wandering away from him during battle. He knows it’s not entirely fair or realistic that he expect them to be at his side the entire time, but it relaxes him in a way… He can’t concentrate when his s/o isn’t in his line of vision
•So losing them in the commotion makes him panic. He’d probably end up getting distracted from battle and taking a hit from the enemy while he looked around for his s/o. After the battle, once he found them, they would probably scold him for being so foolish while patching up his wound
•Kankuro would be equally furious and wrecked to see his s/o die in battle, but he’s the one who starts crying the minute he sees their dead body. He probably holds them as they die, or doesn’t let go of their body. At that just makes it more traumatizing.
•Kankuro visits their grave a lot and asks for forgiveness. He doesn’t know why. He just always felt like there was something he could have done to save them, so he kneels down at the grave and tells them he’s sorry. He’s so sorry and he misses them. He has to come to their grave alone though, because he cries a lot. Even years down the line, a few tears will slip out as he visits the altar