We discuss a natural form of Ricci-Flow conjugation between two distinct general relativistic data sets given on a compact $n$-dimensional manifold. The Ricci flow generates a form of $L^2$ parabolic averaging, of one data set with respect to the other, with a number of desiderable properties: (i) Preservation of the dominant energy condition; (ii) Localization by a heat kernel, (associated with the linearized Ricci flow), whose support sets the scale of averaging; (iii) Entropic stability.