We examine galaxy groups from the present epoch to z ∼ 1 to explore the impact of group dynamics on galaxy evolution. We use group catalogues from the Sloan Digital Sky Survey (SDSS), the Group Environment and Evolution Collaboration (GEEC) and the high-redshift GEEC2 samples to study how the observed member properties depend on the galaxy stellar mass, group dynamical mass and dynamical state of the host group. We find a strong correlation between the fraction of non-star-forming (quiescent) galaxies and galaxy stellar mass, but do not detect a significant difference in the quiescent fraction with group dynamical mass, within our sample halo mass range of ∼1013–1014.5 M⊙, or with dynamical state. However, at z ∼ 0.4 we do find some evidence that the quiescent fraction in low-mass galaxies [log10(Mstar/M⊙) ≲ 10.5] is lower in groups with substructure. Additionally, our results show that the fraction of groups with non-Gaussian velocity distributions increases with redshift to z ∼ 0.4, while the amount of detected substructure remains constant to z ∼ 1. Based on these results, we conclude that for massive galaxies [log10(Mstar/M⊙) ≳ 10.5], evolution is most strongly correlated to the stellar mass of a galaxy with little or no additional effect related to either the group dynamical mass or the dynamical state. For low-mass galaxies, we do find some evidence of a correlation between the quiescent fraction and the amount of detected substructure, highlighting the need to probe further down the stellar mass function to elucidate the role of environment in galaxy evolution.