We develop and implement an algorithm for the quantitative characterization of cluster dynamics occurring for cellular automata defined on an arbitrary structure. As a prototype for such systems we focus on the Ising model on a finite Sierpinski gasket, which is known to possess a complex thermodynamic behavior. Our algorithm requires the projection of evolving configurations into an appropriate partition space, where an information-based metric (the Rohlin distance) can be naturally defined and worked out in order to detect the changing and the stable components of clusters. The analysis highlights the existence of different temperature regimes according to the size and the rate of change of clusters. Such regimes are, in turn, related to the correlation length and the emerging 'critical' fluctuations, in agreement with previous thermodynamic analysis, hence providing a non-trivial geometric description of the peculiar critical-like behavior exhibited by the system. Moreover, at high temperatures, we highlight the existence of different timescales controlling the evolution towards chaos.