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For an infinite discrete group $G$ acting on a compact metric space $X$, weintroduce several weak versions of equicontinuity along subsets of $G$ and showthat if a minimal system $(X,G)$ admits an invariant measure then $(X,G)$ isdistal if and only if it is pairwise IP$^*$-equicontinuous; if the productsystem $(X\times X,G)$ of a minimal system $(X,G)$ has a dense set of minimalpoints, then $(X,G)$ is distal if and only if it is pairwiseIP$^*$-equicontinuous if and only if it is pairwise central$^*$-equicontinuous;if $(X,G)$ is a minimal system with $G$ being abelian, then $(X,G)$ is a systemof order $\infty$ if and only if it is pairwise FIP$^*$-equicontinuous.

Characterizations of distality via weak equicontinuity