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Is what I thought after this chapter
It is the general idea that Nagato became Pain during the 3rd war, and this is also the case:
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Here we see Nagato telling naruto that suna, iwa and konoha were in the middle of war.
However, when the Kannabi bridge was destroyed, the war was said to end:
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Of course this is told as a "story-teller-perspective", and we have no knowledge how much time they waited before destroying the bridge. So Kishi could still explain it that way, though it would be a bit lame (IMO).
So we can conclude is the general expectation that the war ended after this, but yet Kishi wrote that when Yahiko was killed they were still in the middle of war.
So it all depends on how long Obito spend in Madaras cave. Also we have no actual proof about how long time there is between the destruction of the bridge and Narutos birth. Most people have a general belief that it is 1 year, which is why all people questioned Obito to grow so much, become stronger etc. in just a year.
Perhaps it wasn't a 1 year? Clearly what we see now, must tell us that it wasn't. So it is possible that Kishi wants to change this by showing us that the war actually didn't end straight after the destruction of the bridge, or that they took a lot more time to destroy the bridge. There is no actual evidence about when the bridge was destroyed. And when Kakashi was ambushed by these mist hunters he looked just like he did, when he waked up after being saved by Minato:
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Except he wears arm guards on both arms. But that isn't the point. Minato might have given them another mission meanwhile and then Minato went somewhere on his own. So it could be explained this way. Perhaps Kakashi needed to heal his injuries for some time as well.
I must say though this explanation could work, it seems a lot like Kishi didn't think it through back at Kakashis gaiden, of course, but come on, we can't expect that much from the man U_U