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During the COVID-19 pandemic, a major driver of new surges has been the emergence of new variants. When a new variant emerges in one or more countries, other nations monitor its spread in preparation for its potential arrival. The impact of the new variant and the timings of epidemic peaks in a country highly depend on when the variant arrives. The current methods for predicting the spread of new variants rely on statistical modeling, however, these methods work only when the new variant has already arrived in the region of interest and has a significant prevalence. Can we predict when a variant existing elsewhere will arrive in a given region? To address this question, we propose a variant-dynamics-informed Graph Neural Network (GNN) approach. First, we derive the dynamics of variant prevalence across pairs of regions (countries) that apply to a large class of epidemic models. The dynamics motivate the introduction of certain features in the GNN. We demonstrate that our proposed dynamics-informed GNN outperforms all the baselines, including the currently pervasive framework of Physics-Informed Neural Networks (PINNs). To advance research in this area, we introduce a benchmarking tool to assess a user-defined model's prediction performance across 87 countries and 36 variants.