  Experiments, theory and atomistic simulations show that finite triple
junction mobility results in non-equilibrium triple junction angles in evolving
polycrystalline systems. These angles have been predicted and verified for
cases where grain boundary migration is steady-state. Yet, steady-state never
occurs during the evolution of polycrystalline microstructures as a result of
changing grain size and topological events (e.g., grain face/edge switching -
"$T_1$" process, or grain disappearance "$T_2$" or "$T_3$" processes). We
examine the non-steady evolution of the triple junction angle in the vicinity
of topological events and show that large deviations from equilibrium and/or
steady-state angles occur. We analyze the characteristic relaxation time of
triple junction angles $\tau$ by consideration of a pair of topological events,
beginning from steady-state migration. Using numerical results and theoretical
analysis we predict how the triple junction angle varies with time and how
$\tau$ varies with triple junction mobility. We argue that it is precisely
those cases where grain boundaries are moving quickly (e.g., topological
process in nanocrystalline materials), that the classical steady-state
prediction of the finite triple junction mobility triple junction angle is
inapplicable and may only be applied qualitatively.
