Dynamic real-world networks, encompassing both digital and physical realms, inherently display complex spatio-temporal phenomena. A common manifestation is the propagation of node states, where information is disseminated through network edges via everyday human and system interactions. Given the potential threats like virus spread and fake news dissemination, it is critical to quickly and effectively identify propagation patterns and their harmful instances. Although various approaches exist for classifying spatio-temporal graphs, we argue that current methods overlook essential characteristics of propagation behavior, such as the causal-effect relationships between node state transitions. To address this gap, we propose a novel cross-snapshot attention method that leverages the unique features of propagations originating from specific nodes over time. The novelty lies in the element-wise attention weight calculations across consecutive snapshots, linking changes in propagation states to local network regions. Our method surpasses a set of established graph neural network techniques in accuracy across datasets designed to simulate complex real-world propagation dynamics. We performed a series of ablation studies to confirm the positive impact of the cross-snapshot attention module and its robustness to missing snapshots, which shows that our method experiences smoother performance degradation compared to the state-of-the-art.