The relationships between diffusivity and the excess, pair and residual multiparticle contributions to the entropy are examined for Lennard-Jones liquids and binary glassformers, in the context of approximate inverse power law mappings of simple liquids. In the dense liquid where diffusivities are controlled by collisions and cage relaxations, Rosenfeld-type excess entropy scaling of diffusivities is found to hold for both crystallizing as well as vitrifying liquids. The crucial differences between the two categories of liquids emerge only when local cooperative effects in the dynamics result in significant caging effects in the time-dependent behaviour of the single-particle mean square displacement. In the case of glassformers, onset of such local cooperativity coincides with onset of deviations from Rosenfeld-type excess entropy scaling of diffusivities and increasing spatiotemporal heterogeneity. In contrast, for two- and three-dimensional liquids with a propensity to crystallise, the onset of local cooperative dynamics is sufficient to trigger crystallization provided that the liquid is sufficiently supercooled that the free energy barrier to nucleation of the solid phase is negligible. The state points corresponding to onset of transient caging effects can be associated with typical values, within reasonable bounds, of the excess, pair, and residual multiparticle entropy as a consequence of the isomorph-invariant character of the excess entropy, diffusivity and related static and dynamic correlation functions.