Graph-based methods pervade the inference toolkits of numerous disciplines including sociology, biology, neuroscience, physics, chemistry, and engineering. A challenging problem encountered in this context pertains to determining the attributes of a set of vertices given those of another subset at possibly dierent time instants. Leveraging spatiotemporal dynamics and prior information can drastically reduce the number of observed vertices, and hence the cost of sampling. Alleviating the limited exibility of existing approaches, this thesis broadens the kernel-based graph function estimation framework to reconstruct time-evolving functions over possibly time-evolving topologies. This encompassing approach inherits the versatility and generality of kernel-based methods, for which no knowledge on distributions or second-order statistics is required. Ecient inference algorithms are derived that operate in an online and even data-adaptive fashion. Moreover, semi-parametric approaches capable of incorporating the structure of known graph functions without sacri- cing the exibility of the overall model are advocated. Numerical tests with real data sets corroborate the merits of the proposed methods relative to competing alternatives.