We perform a detailed study of the interdiffusion of two miscible liquids of different viscosities, water and a water/glycerol mixture, flowing side by side within a pressure-driven microfluidic flow. Using Raman imaging, we measure cartographies of the concentration in glycerol for different imposed flow rates. The analysis of these experimental data confirms a classical diffusive mixing since the extent of the diffusion layer follows a 1/2 power law regime with the streamwise coordinate x. However, the interdiffusion layer is not located in the center of the microchannel, and its mean position ym evolves with x during the mixing. We also derive a theoretical model and present analytical arguments to explain the above results. We demonstrate that the displacement of the interdiffusion zone ym(x) is due to the coupling between hydrodynamics and the mixing through the dependence of the viscosity with the glycerol concentration. Eventually, we perform numerical simulations of the theoretical model and compare the solutions with the experimental data in order to estimate quantitatively an interdiffusion coefficient.