Network alignment, which maps the same entities across multiple networks, has gained tremendous interest in recent years. However, most existing alignment methods propose to align static graphs that are merely a single snapshot of real-world networks. These methods fail to model the inherent dynamics of entire networks where the nodes, links, and attributes are bound to change over time, making dynamic network alignment challenging. Moreover, modeling the interaction between two different dynamic graphs comes with additional challenges: (1) catastrophic forgetting while learning the evolution of individual networks, (2) distributional shift on the same dynamic network, and (3) domain differences between the two networks. Hence, to overcome these challenges, we propose RoCoNA, an end-to-end reservoir sampling-based continual learning approach built over streaming Graph Neural Networks (GNNs) that uses a novel shift-induced regularizer to handle distribution drift and domain differences in evolving networks. We empirically show that our method outperforms the existing state-of-the-art static and dynamic alignment methods. We perform case studies on networks with high distributional shifts to strongly validate our claims. Code is available at https://github.com/shruti400/RoCoNA/tree/main