Short: yes, it is possible to transport mixtures.
If the pump can process both of the liquids the flow rate may change due to viscosity of the liquid currently inside the pump. For instance, if a succession of water-gas-water-gas volumes with one pump setting (amplitude and frequency) is wanted, the flow rate will increase with the gas phase and drop back with the liquid phase. Same will happen with two differently viscous liquids, though not as drastic as with air-liquid.However, when the separating front of the two media goes into the pump it will be blurred a bit when it comes out again. This means the separation between the two media will not be a sharp front, but a kind of mixture section in between. With air-water there will be some foamed section (which may collapse after some travelling along the tube line), with two liquids, there will be a real mixture (example: a succession of red and blue water entering the pump will come out as violet in this mixture section). How large this mixture section will be or if it will be there at all has to be tested, as this is also dependent on the pump settings and flow rate.