The dynamics of cellular automata running on cyclic configurations under block-sequential deterministic updates may drastically depend on the update employed. In order to better understand such a dependence, various analyses have been carried out in the literature. In the same direction, here we explore the potential of a recently proposed metric meant to act as a measure of the asynchrony degree of an update; we emphasise that this the first effort trying to probe the potential of the metric as it relates to the dynamics of automata networks. More specifically, we investigate, for all elementary cellular automata, when update schedules with the same degree of asynchrony have similar dynamics, as expressed by their leading to structurally equivalent basins of attraction, that is, those with the same attractors and same configurations in each one. Considering all configuration sizes from 5 to 13, we performed a computational analysis of the dynamics of the rules, trying to correlate them with the asynchrony degree of the update schedules, and grouping the rules into similarity classes of their dynamics. Due to computational limitations, the correlation between the asynchronous degrees and the dynamics sought was not entirely conclusive, but was clearly indicative of the relevance of the metric and the methodology employed.