It is possible to derive both Cahn-Hilliard (CH) and Cahn-Allen (CA) equations using variational arguements and steepest-descent principles. In this sense, there is an unifying mathematical basis for both these equations of microstructural evolution - And, the following paper discusses such variational methods in great detail:
The idea behind these variational principles, as described in the above reference, seems to be the following: Thermodynamics, provides us with quantities that either monotonically increase or decrease (say, entropy and free energy). However, to know the time rate at which these quantities change - in other words, the dynamics - we need some external principles. The concept of gradient flow based on inner products is used as such an external principle to derive the CH and CA equations from the corresponding free-energy functionals. The inner product in this case indicates the kinetic distance between two microstructures and gradient flow chooses the direction in which the system can move to decrease the corresponding free energy as fast as possible.