1. The α, β, γ diversity decomposition methodology is commonly used to investigate changes in diversity over space or time but rarely conjointly. However, with the ever-increasing availability of large-scale biodiversity monitoring data, there is a need for a sound methodology capable of simultaneously accounting for spatial and temporal changes in diversity. 2. Using the properties of Chao's index, we adapted Rao's framework of diversity decomposition between orthogonal dimensions to a multiplicative α, β, γ decomposition of functional or phylogenetic diversity over space and time, thereby combining their respective properties. We also developed guidelines for interpreting both temporal and spatial β-diversities and their interaction. 3. We characterised the range of β-diversity estimates and their relationship to the nested decomposition of diversity. Using simulations, we empirically demonstrated that temporal and spatial β-diversities are independent from each other and from α and γ-diversities when the study design is balanced, but not otherwise. Furthermore, we showed that the interaction term between the temporal and the spatial β-diversities lacked such properties. 4. We illustrated our methodology with a case study of the spatio-temporal dynamics of functional diversity in bird assemblages in four regions of France. Based on these data, our method makes it possible to discriminate between regions experiencing different diversity changes in time. Our methodology may therefore be valuable for comparing diversity changes over space and time using large-scale datasets of repeated surveys.