This contribution is concerned with the fundamental thermodynamic aspects of solid-fluid phase transformations in stressed rocks, specifically in the context of pressure solution. We concentrate in particular on the formulation of a kinetic law governing the migration of stressed and curved solid-fluid phase boundaries, an objective that is achieved by using the methods of the thermodynamics of irreversible processes. We then apply our result to the study of the geometrical evolution of a fluid-filled cylindrical pore embedded in an isotropic, linear elastic solid that is subject to a hydrostatic remote stress, assuming that the interface kinetics controls the phase boundary migration and allowing for the effects of capillarity. On the basis of this investigation, we obtain an analytical expression for the pore's growth and show that phase equilibrium along the cylindrical solid-fluid phase boundary is possible only when the pore pressure exceeds a critical value. The phase equilibrium is found to be kinetically unstable: when subjected to a small perturbation of its radius, the pore will either grow or shrink. The nature of this instability is further explored in the companion paper.