When Kakashi compares his relationship to Obito to the connection that Naruto and Sasuke share, Kakashi shows that he admires Naruto for always striving to save Sasuke and repeatedly reaching out to his friend. Kakashi notes that Naruto is stronger than him and possesses a resolve to save his friends that will never waver, unlike Kakashi, who always seems unable to save his friends. In the end, the only thing Kakashi knows to do to deal with his fallen friends is to kill them, protecting the memory of their past selves by killing their current selves.

Kakashi's approach seems to be a coping mechanism. He can't acknowledge that his idol Obito could have fallen so far and the kind-hearted, heroic boy he once knew could be the villainous mass-murderer Obito had become. I think it's same for Sasuke, which is why Kakashi referred to Sasuke as brainwashed instead of acknowledging his decisions as made of Sasuke's own volition and free will. 

Kakashi is shocked when he first catches a glimpse of Sasuke fighting alongside Naruto, having no explanation of why Sasuke is there and apparently now an unexpected ally.

Later, when Kakashi emerges from his Kamui dimension and has his first wartime run-in with Sasuke, Sasuke is absolutely disinterested in Kakashi, solely concerned about eliminating Obito. Sasuke barely registers Kakashi's emergence, treating his sensei with pure indifference. The only emotion is surprise that Kakashi is randomly popping up out of the Kamui dimension, though this time he does acknowledge him by name. 

Kakashi is a little more reactionary, as Kakashi promises to talk with Sasuke later, though I doubt that Sasuke is interested or cares to talk to Kakashi. I don't think Sasuke has anything to say to Kakashi, though it appears Kakashi has something in mind he wishes to say to Sasuke later. Perhaps Kakashi wants to catch up to understand why Sasuke, previously crazy, brainwashed, murderous, and bent on destroying Konoha, is now suddenly is an ally to the five nations. Considering how Kakashi has only previously witnessed teen Sasuke behaving like an unhinged psychopath, it's unusual that Kakashi is so calm about Sasuke's presence on the battlefield, taking it in stride.

I find it interesting that Kakashi apologizes to Sasuke for randomly appearing out of the blue. Despite being focused on the current task of killing Obito, Kakashi addresses and acknowledges Sasuke, which not even Naruto gets in this scene.

Kakashi probably accrued a new interest in saving Sasuke now that Kakashi understands the pain Naruto feels, the pain of having a best friend be a villain and the desperate need to redeem a former friend. Kakashi has been disinterested in helping Sasuke before, but reuniting with Obito made Kakashi empathize with Naruto and Sakura in ways he hadn't before. For the first time, Kakashi shows some empathy and consideration towards his team.

He expresses genuine sadness and empathy towards Naruto for the incredible resolve it takes to never give up on a fallen friend. Additionally, this war is the first and only time Kakashi cares about Sakura's feelings for Sasuke, as he admires her persevering kindness towards the man she loves. He treats Sakura's enduring love for Sasuke with a newfound admiration, marveling how admirably kind and gentle she is. Kakashi even stands up for Sakura against Sasuke a couple times during the war, and. Kakashi even treats Sasuke more tactfully for a change.

Kakashi experiences a lot of personal growth during the war arc, and Kakashi's wartime moments are pretty much the only time that Kakashi actually treats his team well, treating them with respect and empathy that have rarely, if ever, been there before.

Of course, Kakashi's burst of maturity towards his old team is short-lived and gone just as quickly as it emerged after they defeat Kaguya. But for a few scenes, Kakashi actually considers his team's feelings and dreams is nice to see while it lasts.

After having his eye stolen by Madara, Kakashi waxes nostalgic as he watches his former team plan and bicker, united once again after 4 years. Remembering how young and immature they once were, Kakashi muses how far they have each come, recalling the team's introductions and their dreams.

When it comes to Sasuke, Kakashi notes that Sasuke achieved his goal by killing Itachi, but Kakashi wonders what it is that Sasuke desires next, what is his motivation for being there in the war?

Kakashi internally admits he made the mistake of believing he understood everything about Sasuke, but now he has the maturity to realize that he understood anything at all about Sasuke, and he never did, and he still doesn't, and he's completely in the dark as to Sasuke's motivations.

For the moment, Kakashi is considerate of the unique individual he is speaking to, treading carefully and respectfully around Sasuke.

Curiously, Kakashi asks Sasuke what his goal currently is. I appreciate the prudence of Kakashi's query, since Sasuke's sudden appearance as an unexpected ally without explanation is alarming and suspicious. Making it even worse is that even when directly questioned about his motivations by Kakashi, Sasuke says nothing, making him exponentially more dubious and untrustworthy. Since Sasuke had freely explained his intentions to destroy Konoha, which was bad enough in itself, that makes Sasuke being cagey about his current objective is concerning to say the least. 

Rather than push the issue, Kakashi calmly backs down and gives Sasuke space instead by saying that Sasuke doesn't have to tell anything right now if he doesn't want to. I don't know that I would be that lenient, since Sasuke's silence is alarming and worrisome. Naruto and Sakura are worried about it too but too preoccupied with more pressing matters to raise a fuss about it. Besides, none of them are in any position to refuse Sasuke's help, and doing anything to aggravate or incense a powerful ally like Sasuke may be a luxury the team can't afford.

Of course, the team might just be aware of Sasuke's extreme stubbornness and that pushing him to talk when he's chosen not to will do little good. Perhaps that's why Kakashi backs off, assuming that grilling Sasuke on his motives will get Kakashi nowhere. Aside from that, the team is supposed to be battle ready, so they don't exactly have time to chit-chat about motivations when the answer isn't readily forthcoming from Sasuke. Sasuke's stony silence made it clear that he doesn't wish to share anything with Team 7. A rather interesting turnaround from last time, since Sasuke was openly spewing his feelings out to Team 7. Yet now that Kakashi actually wants to understand and hear Sasuke's motivations and feelings, Sasuke is silent.

Sasuke is difficult to read, since he's especially stoic and keeping his emotions boxed off and tightly under control. It's hard to say how much of his Revolution plan Sasuke has come up with by this point, as his plan seemed to be ever-shifting during the war and he was hashing out a plan as the fight goes on (which isn't the easiest thing, since the fight itself has to be taking up a considerable amount of concentration, leaving Sasuke with little time or space to really come up with a plan for world peace in the meantime). Sasuke's refusal to share information is because he knows Team 7 won't like what they hear, Sasuke has no interest in Kakashi anymore, and Sasuke probably doesn't yet fully know himself what he wants to do, as his purpose in the flux, fluid and everchanging while Sasuke seeks an idea for world peace, turning over in his mind possible ideas that work versus what won't, pondering how to create a world peace plan that will work and be sustainable long-term.

Sasuke has essentially completely cut all emotional ties with Kakashi by this point. All fury or affection towards Kakashi is completely gone, replaced by cool indifference. Sasuke simply doesn't care about Kakashi anymore on any personal level. Kakashi has sunk to a level of acquaintance, no longer having any personal sway, influence, or impact over Sasuke more than any random other person does.

As for Kakashi, he's very lenient and patient here, surprisingly so. This time, Kakashi asks Sasuke what he wants and feels instead of telling Sasuke what he should want and what he should feel.

Unlike in the past, Kakashi doesn't launch into a preachy lecture. Rather, Kakashi expresses genuine curiosity in understanding Sasuke's motivations and feelings, which signals to Sasuke that if he wants to talk about his feelings, he has a willing listener interested in what he has to say. This marks the first time Kakashi has expressed such sentiment, as previously Kakashi had only shut down Sasuke's feelings and encouraged Sasuke to suppress his emotions because it was inconvenient for others to deal with them.

Kakashi mentions that he doesn't think it is insignificant or meaningless that fate brought Team 7 together again, expressing hopeful optimism for the team's reformation. Kakashi waxes nostalgic, perhaps not realizing how much he'd missed having his team whole again, now that they are cooperating together as a team again.

Kakashi learns from Sakura that apparently Sasuke voiced his intention to become Hokage. It's hard to guess what Kakashi thinks about this, since Kakashi doesn't react to that startling news much. Kakashi wouldn't be able to guess that Sasuke's version of a Hokage is radically different from what the position has traditionally been in the past, so it's hard to say whether Kakashi finds this sudden turnaround extremely alarming or if Kakashi thinks it's a positive sign that Sasuke wants to return to the village and even intends to become village leader. I'm not sure what's going through Kakashi's head on that tidbit of info. Maybe Kakashi himself is too overwhelmed by the strange turn of events, so he decides to save concerns about Sasuke for later and focus on the battle first.

Kakashi reminds his former students of his first lesson on teamwork. From Sasuke's scoff, I'd guess Sasuke isn't appreciative of Kakashi still pretending to be Sasuke's sensei, though Sasuke doesn't loudly protest either, he just doesn't look especially pleased by Kakashi's reminder of their team days.

Later on in the war, we see that Kakashi is willing to go so far as even sacrificing his life to save Sasuke, recognizing that Sasuke is more crucial to winning the war than himself.

As for Kakashi's Susano'o, Sasuke's reaction was shocked disbelief as he wondered how such a thing could be possible since Kakashi no longer had his Sharingan, so it should be impossible for Kakashi to possess such a jutsu. Judging by Sasuke's comments, Sasuke recognized the chakra as Obito's and wondered how Obito's chakra could return when Obito himself was already dead. Well, that Susano'o was a bit strange and nontypical, so no wonder Sasuke is shocked by it.