We introduce a new framework to analyze shape descriptors that capture the geometric features of an ensemble of point clouds. At the core of our approach is the point of view that the data arises as sampled recordings from a metric space-valued stochastic process, possibly of nonstationary nature, thereby integrating geometric data analysis into the realm of functional time series analysis. Our framework allows for natural incorporation of spatial-temporal dynamics, heterogeneous sampling, and the study of convergence rates. Further, we derive complete invariants for classes of metric space-valued stochastic processes in the spirit of Gromov, and relate these invariants to so-called ball volume processes. Under mild dependence conditions, a weak invariance principle in $D([0,1]\times [0,\mathscr{R}])$ is established for sequential empirical versions of the latter, assuming the probabilistic structure possibly changes over time. Finally, we use this result to introduce novel test statistics for topological change, which are distribution-free in the limit under the hypothesis of stationarity. We explore these test statistics on time series of single-cell mRNA expression data, using shape descriptors coming from topological data analysis.