We investigate the properties of the velocity gradient tensor for spatially evolving turbulent flows (a near-wake, two axisymmetric jets and a planar mixing layer). Emphasis is placed on the study of the normal and non-normal parts of the tensor. Non-normality plays a greater role in the dynamics than is observed for HIT and does so for all spatial locations examined. This implies a greater role for the deviatoric part of the pressure Hessian. Results for the wake flow, where we isolate the coherent part of the dynamics using a modal decomposition, clarify how these competing effects operate. Previous studies have shown the shape of the Q–R diagram (formed by the second and third invariants of the characteristic equation for the tensor) is approximately universal at small-scales for different flows. The non-normal dynamics are neglected in the Q–R approach but appear to differ significantly between flows.