The style of melt migration in the mantle is important to the interpretation of basalts erupted on the surface. Both grain-scale diffuse porous flow and channelized melt migration have been proposed. Through high-order accurate numerical simulations, we show that strong nonlinear interactions between compaction and dissolution in an upwelling mantle give rise to porosity waves and high-porosity melt channels that have well organized but time-dependent structures. Only the upper part of the channel is pyroxene-free dunite. The lower part is harzburgite. Transient melt flow in the wave regime results in significant lateral mixing and chromatographic fractionation even when mantle source compositions are independent of time. Caution must be exercised when inferring the geometry and spatial distribution of mantle heterogeneity based on spatial and temporal variations in isotopic ratios recorded in basalts.