We investigate the enstrophy dynamics in relation to objective Eulerian coherent structures (OECSs) and their impact on the enstrophy and scalar transport near the turbulent/non-turbulent interface (TNTI) in flows with and without stable stratification. We confirm that vortex-stretching produces enstrophy inside the boundaries of the OECSs, while viscous diffusion transfers the enstrophy across the boundaries of the structures. Although often overlooked in the literature, viscous dissipation of enstrophy within the boundaries of vortical structures is significant. Conversely, for the weakly stratified flows also investigated here, the effect of the baroclinic torque is negligible. We provide evidence that the OECSs advect the passive/active scalar and redistribute it via molecular diffusion. Finally, we use conditional analysis to show that the typical profiles of the enstrophy and scalar transport equation terms across the TNTI are compatible with the presence of OECSs positioned at the edge between the turbulent sublayer and the turbulent core region. We show that when these profiles are further conditioned to the presence of OECSs, their magnitude is considerably higher.