In an unsteady pulsatile non-Newtonian fluid past a tube with a thin wall layer, the dispersion of a narrow uniform slug of injected solute over a cross-section is examined. At the interface between the mobile fluid phase and the immobile wall phase, both irreversible and reversible reactions have been adopted. The Carreau-Yasuda model is used to describe the fluid's rheology. The impacts of fluid rheology and reaction parameters on the concentration profiles in the fluid-and wall-phases and the three transport coefficients, viz, the depletion coefficient (K0), the convection coefficient (K1), the dispersion coefficient (K2) in the fluid phase are predicted numerically. A considerable shift in the behaviour of K1 and K2 with a higher reaction rate may be observed in the transient stage. The axial dispersion of mobile-phase concentration in the unsteady Carreau-Yasuda II fluid model is significantly larger than in Poiseuille and steady Carreau-Yasuda II fluid models, and flow pulsatility on the immobile-phase concentration is prominent upstream at a longer time. In addition, the peak value of the mobile-phase section-mean concentration is consistently lower than in other fluid models. This study could help researchers to understand the drug delivery in blood vessels and pulmonary mechanical ventilation. © 2022 The Author(s).