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Recently there has been significant interest in the analysis of finite-blocklength performance in different settings. Specifically, there is an effort to extend the performance bounds, as well as the Gaussian approximation (dispersion) beyond point-to-point settings. This proves to be a difficult task, as the performance may be governed by multiple dependent constraints. In this work we shed light on these difficulties, using the multiple-access channel as a test case. We show that a local notion of dispersion is more informative than that of dispersion regions sought after thus far. On the positive side, we show that for channels possessing certain symmetry, the dispersion problem reduces to the single-user one. Furthermore, for such channels, linear codes enable to translate single-user achievability bounds to the multiple-access channel.