Melting of a plume head can affect its dynamics by creating melt retention buoyancy. Melt migration controls the distribution of the melt retention buoyancy and affects the dynamics of the plume. We investigate in detail the effects of melt migration on the dynamics and partial melting of a 20-km radius mantle diapir, using axisymmetric two-phase flow models. We first study melt migration in a diapir with a 10% initial melt where no further melting is allowed. The diapir dynamics are modeled for permeable and impermeable cases. In the permeable model, melt migrates within the diapir, whereas no relative motion is allowed between melt and solid matrix in the impermeable model. The permeable model shows a progressive increase of melt fraction in the top portion and decrease in the bottom part of the diapir. This results in a melt buoyancy polarization that elongates the diapir and increases its upwelling velocity. We then model the dynamics of the permeable and impermeable diapirs allowing melting to occur. The velocity of the permeable diapir and its melt generation are significantly larger than that of the impermeable diapir. In general, the permeable diapirs lack the mushroom shape observed for the impermeable ones. Because of intensive computational demand of the two-phase flow modeling, our detailed studies are limited to small diapirs. However, we also investigate a 200-km-diameter mantle plume and show that melt migration produces a nibble of higher melt fraction at the top of the plume that ascends much faster than the bulk of the plume.