  We investigate four-dimensional, self-dual gravitational instantons endowed
with a product structure RxM_3, where M_3 is homogeneous of Bianchi type. We
analyze the general conditions under which Euclidean-time evolution in the
gravitational instanton can be identified with a geometric flow of a metric on
M_3. This includes both unimodular and non-unimodular groups, and the
corresponding geometric flow is a general Ricci plus Yang-Mills flow
accompanied by a diffeomorphism.
