Optical coherence tomography (OCT) can monitor human donor corneas non-invasively during the de-swelling process following storage for corneal transplantation, but currently only resultant thickness as a function of time is extracted. To visualize and quantify the mechanism of de-swelling, we present a method exploiting the nanometer sensitivity of the Fourier phase in OCT data to image deformation velocities. The technique was demonstrated by non-invasively showing during de-swelling that osmotic flow through an intact epithelium is negligible and removing the endothelium approximately doubled the initial flow at that interface. The increased functional data further enabled the validation of a mathematical model of the cornea. Included is an efficient method of measuring high temporal resolution (1 minute demonstrated) corneal thickness, using automated collection and semi-automated graph search segmentation. These methods expand OCT capabilities to measure volume change processes for tissues and materials.