I agree with much of what the answer before me has already expressed—we're not really given all that much information as to the nature of the war which preceded the events of Much Ado About Nothing.
It's all very vague (which does make sense, given the constraints of drama). One can have expositional dialogue, perhaps, but exposition via dialogue will always be, by its very nature, more unwieldy than what can be achieved in a work of prose-fiction. More information could have been more clearly expressed, but, at the same time, one might wonder whether the actual details of the conflict were ever meant to be in the foreground begin with.
They were at war, and one can assume that Don John very much seems to have played a part in these events (though, as Tamara K. H. points out, one should not be too confident about John's role in the war, given that he's still standing around in the aftermath).
In any event, it might be worth adding that Shakespeare's plays tend to be very much open to...
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