arXiv:2610.25433v1 Announce Type: cross Abstract: Longitudinal dispersion in a yielding fluid depends on the velocity distribution and on how the flow is maintained. We study a neutral tracer in pressure-assisted electrokinetic and magnetohydrodynamic flow of a Casson fluid through a slit microchannel. The nonlinear stress and velocity equations determine the yielded layers, the unyielded core and the axial flow. Mei's multiple-scale homogenization of the advection-diffusion equation then separates transverse relaxation, mean drift and longitudinal spreading, yielding the Taylor-Aris auxiliary problem and effective diffusivity. An exact pressure-driven solution gives cubic decay of the mean velocity and eighth-power decay of the dispersion enhancement near flow arrest. Under fixed pressure and electrical forcing, increasing the yield parameter from $Y=0$ to $0.7$ moves the optimum Hartmann number from $Ha_p=0.696$ to $2.634$, while the corresponding normalized diffusivity decreases from $D_T^{\mathrm C}=1.454$ to $1.015$. At fixed mean throughput and $Ha_p=1$, the trend reverses: $D_T^{\mathrm C}$ increases from $1.000433$ to $1.002311$. The homogenized energy identity traces this difference to the velocity departure from the sectional mean, including that within the unyielded core. Analytical limiting solutions, direct concentration calculations, numerical convergence studies and a coupled Casson shear test support the reduced-flow predictions. An electrolyte-based dimensional assessment identifies the magnetic optima as theoretical forcing states for representative aqueous microchannels. These results are relevant to predicting longitudinal tracer spreading in microchannels carrying fluids with a yield stress, whether pressure or throughput is controlled.