  Two-dimensional (2D) particulate aggregates formed due to competing
interactions exhibit a range of non-equilibrium steady state morphologies from
finite-size compact crystalline structures to non-compact string-like
conformations. We report a transition in heterogeneous microscopic dynamics
across this morphological hierarchy as a function of decreasing long-range
repulsion relative to short-range attraction at a constant {\it low} density
and temperature. Following a very slow cooling protocol to form steady state
aggregates, we show that geometric frustration inherent to competing
interactions assures non-ergodicity of the system, which in turn results in
long-time sub diffusive relaxation of the same. Analysing individual particle
trajectories generated by molecular dynamics, we identify {\it caging} dynamics
of particles in compact clusters in contrast to the {\it bonding} scenario for
non-compact ones. Finally, by monitoring temperature dependence, we present a
generic relation between diffusivity and structural randomness of the
aggregates, irrespective of their thermodynamic equilibrium.
