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The problem of sequential change diagnosis is considered, where a sequence of independent random elements is accessed sequentially, there is an abrupt change in its distribution at some unknown time, and there are two main operational goals: to quickly detect the change and to accurately identify the post-change distribution among a finite set of alternatives. A standard algorithm is considered, which does not explicitly address the isolation task and raises an alarm as soon as the CuSum statistic that corresponds to one of the post-change alternatives exceeds a certain threshold. It is shown that in certain cases, such as the so-called multichannel problem, this algorithm controls the worst-case conditional probability of false isolation and minimizes Lorden’s criterion, for every possible post-change distribution, to a first-order asymptotic approximation as the false alarm rate goes to zero sufficiently faster than the worst-case conditional probability of false isolation. These theoretical results are also illustrated with a numerical study.