  We derive the geodesic equation for relatively K\"ahler metrics on fibrations
and prove that any two such metrics with fibrewise constant scalar curvature
are joined by a unique smooth geodesic. We then show convexity of the log-norm
functional for this setting along geodesics, which yields simple proofs of
Dervan and Sektnan's uniqueness result for optimal symplectic connections and a
boundedness result for the log-norm functional. Next, we associate to a
fibration degeneration a unique geodesic ray defined on a dense open subset.
Calculating the limiting slope of the log-norm functional along a globally
defined smooth geodesic ray, we prove that fibrations admitting optimal
symplectic connections are polystable with respect to a large class of
fibration degenerations that are smooth over the base. We give examples of such
degenerations in the case of projectivised vector bundles and isotrivial
fibrations.
