I'll take a stab.  As a preface, I'll say it's hard for me to know much from what you've written about your experience and perspective, so please forgive me if some (or even most) of what I say is something you already knew.

Non-equilibrium thermodynamics is in general a tough field, lacking a general theory as broadly successful as that of equilibrium.  However, considerable success has come in the regime of small perturbations away from equilibrium -- the linear regime of irreversible thermodynamics.  Here theoretical results by Onsager and Prigogine and others have led to a consistent and successful theory.  It is this linear theory upon which the vast majority of applications of phase-field modeling are based.

The minimization of the free energy functional that you refer to is the equilibrium part of the theory.  For conserved quantities, equations describing transport can be derived through appeal to an associated chemical potential, which follows from the appropriate variational derivative of the functional.  A driving force for transport arises from the gradient of this chemical potential, with the flux linearly proportional to it.  (See Cahn and Hilliard, 1961.)

For non-conserved quantities such as an order parameter, the linear theory is brought to bear by postulating that its time derivative is linearly proportional to the variational derivative of the functional with respect to the order parameter.  (Allen and Cahn, 1978.)

So there is a considerable theoretical legitimacy attached to phase-field theory through its connection to the linear theory of non-equilibrium thermodynamics.  Beyond this, phase-field theory proved itself early on by its successful modeling of the kinetics of spinodal decomposition and dendritic solidification in metals.

Part of your question seems to refer to the fact that phase-field theory is macroscopic as opposed to microscopic.  As to whether or not this invalidates its application in the case of irradiation, I can't really say.  However, "microstructural evolution" is usually a macroscopic phenomenon, and if this is true in your case, I don't see why a macroscopic theory wouldn't work -- indeed, I'd imagine you'd require one.  If there is a chemical or physical process driving the microstructural change that must be described atomistically, then it seems like you have an interesting multi-scale problem!  But plenty of chemical reactions can be described adequately using a mean-field framework (away from critical points) -- perhaps the same is true of irradiation.

As long as you introduce an evolution equation in the phase field (such as the Cahn-Hilliard equation) you deal with non-equilibrium thermodynamics. The free energy in NOT conserved. 

Nucleation (of defects, phases, etc) is the tricky part of phase field modelling: it has to be introduced in a (more or less) artificial manner.

Just to be clear, I didn't say that free energy was conserved: I said that transport equations for conserved quantities can be derived through a chemical potential, which is a functional derivative with respect to that quantity of the total free energy.

As Michael Perez mentioned that, unfortunately at the moment phase field modelling is insufficient for kinetic processes in your proposed topic.

 Thank you all for the clarifications.

As far as I understood, despite of the lack of equilibrium in the case of void and bubble growth under irradiation, the phase field methods are well applicable to the second order phase transitions, but their applicability to the first order transitions is less grounded.

Both the second and the first order phase transitions are defined and treated in the framework of thermodynamics. The major difference between the two types of transitions is the existence of the thermodynamic barrier (>> kT) in the case of first order transitions. Sufficiently large stochastic fluctuations are required for such transitions to occur. Stochastic fluctuations can be also treated thermodynamically through the fluctuation-dissipation theorem. Thermodynamic description of the first order transitions is applicable  for the cases of both the sharp and diffuse interfaces between the ambient phase and a new phase nucleus. For example, in the theory of crystallization the diffuse interface approach is more accurate than the sharp interface approximation, which is valid only when the interface thickness is negligible compared with the nucleus size. Under irradiation conditions, two kinds of "forces" are continuously competing with each other. The first one is the external irradiation, which drives the system away from the equilibrium. The second is the internal thermodynamic force moving the system towards the equilibrium. This force can be described thermodynamically through, for example, the chemical potential. In this description both the diffuse and the sharp interface methodologies can be employed. The major problem with the phase-field in the present state is the correct formulation of the thermodynamic "force" through the corresponding thermodynamic potential.