I am not a guru in solidification theory, but from what I know,
The thermal diffusivity if the crytal is a measure of the rate at which heat is carried from the center of the crystal to the surface of the crystal.
Once this heat energy reaches the surface of the crystal, it has to be conducted out the mold. The rate at which this happens is the diffusivity of the mold.
A low duffision through the mold will make it act like a insulator, locking in all the heat.
A low diffusivity of the crystal will result in very steep temperature gradient within the crystal leading to effects such as coring.
Hope that helps