In view of better understanding the properties of waxy crude oils of great practical importance, we study the rheological behavior evolution of model waxy suspensions during transient shear tests, and its variation with concentration in a wide range. Macroscopic tests show a strong destructuring process with the increasing imposed shear rate (starting from rest), while negligible restructuring is observed during shear rate decrease. MRI (Magnetic Resonance Imaging) velocimetry of the suspension during the same tests allow to observe that, in most of the concentration range tested, the material flows only in a limited thickness close to the inner cylinder, and the thickness of this shear-band decreases for increasing concentration but marginally evolves as the apparent shear rate is widely increased then decreased. This means that the extent of the liquid region is governed by the initial start-up flow conditions, and the apparent viscosity variations during the next steps (at other shear rate values) are not associated with an increase of the flowing region but with the evolution of the structure in the liquid region. For the specific case of the lowest wax content analyzed, the thickness of the unsheared region decreases considerably after shearing at a high shear rate. In this case the torque balance in the sample is capable of overcoming the yield stress of the unsheared region. We also show that despite very different (initial) elastic moduli and yield stresses in the solid regime (a factor of several orders of magnitude over the whole range of concentration), we essentially have the same power-law behavior of the fluid in the shear-band, even when the shear-band thickness is close to the particle size.