In this study, modelling of network-structured material flow is considered through a rounded-corner, hyperbolic 4:1:4 contraction-expansion geometry, under axisymmetric configuration. Three representative constitutive models are adopted to represent networked behaviour and to investigate the flow of wormlike micellar fluids in this context. This includes the modified Bautista-Manero (MBM) model (for base thixotropic properties), some newly proposed micellar models (NM_tau (p) and NM_T; for advanced thixotropic modelling), and the EPPT model (for contrast against non-thixotropic properties). In this configuration, emphasis is placed upon interpretation of flow behaviour for these constitutive models, against their response in simple rheometrical flows. To best determine the factors that contribute to epd prediction, current findings have also been contrasted against those reported earlier in Lpez-Aguilar et al. (J Non-Newtonian Fluid Mech 204:7-21, 2014), for the counterpart abrupt rounded-corner, axisymmetric 4:1:4 contraction-expansion flow.