Transient Taylor dispersion controls how anisotropic microstructures such as fibres and rigid colloids spread through finite microfluidic channels, where carrier flows can be strongly non-Newtonian. Classical Taylor–Aris and generalized Taylor dispersion theories determine the long-time coefficient after transverse homogenisation, while finite devices also depend on the pre-asymptotic build-up. We develop a dominant-mode memory-kernel closure for the all-time streamwise variance in bounded shear. The reduction is anchored by the exact short-time kinematic variance (A[u]) and the long-time Taylor enhancement (_L); when sampled-shear memory is governed by one slow mode, their ratio fixes the relaxation time and yields closed laws for the excess variance and instantaneous enhancement. The framework is validated against an exact spectral solution for isotropic dispersion in plane power-law Poiseuille flow and against Newtonian Brownian-rod data with independently supplied rod generalized Taylor dispersion (rod-GTD) coefficients. It is then applied to a semianalytically closed reduced Brownian-rod transport model in plane power-law channels, in which the carrier rheology enters through the imposed velocity and shear fields and semianalytic preprocessing supplies the effective transport profiles together with the long-time transport anchor. The corresponding reduced anisotropic advection–diffusion equation is integrated by lattice Boltzmann simulations to assess transient moment evolution. Across flow indices and rotational P'eclet numbers, the raw transients reflect alignment and rheology-dependent shear sampling, while the reduced variables collapse onto common master curves. The closure also yields development measures (T_d) and (X_d) that locate Taylor onset in finite channels and provide an all-time description for the semianalytically defined reduced rod transport studied here.