  For an infinite discrete group $G$ acting on a compact metric space $X$, we
introduce several weak versions of equicontinuity along subsets of $G$ and show
that if a minimal system $(X,G)$ admits an invariant measure then $(X,G)$ is
distal if and only if it is pairwise IP$^*$-equicontinuous; if the product
system $(X\times X,G)$ of a minimal system $(X,G)$ has a dense set of minimal
points, then $(X,G)$ is distal if and only if it is pairwise
IP$^*$-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 system
of order $\infty$ if and only if it is pairwise FIP$^*$-equicontinuous.
