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Climate change is driving biodiversity loss, disrupting ecosystem functioning, and altering species distributions. Migratory species, whose range varies across seasons depending on specific climatic conditions, are particularly sensitive to environmental changes and serve as indicators of ecosystem health. However, current species distribution models often fail to capture the temporal dynamics critical for migratory species, limiting their ability to provide accurate future range estimations. In this study, we address this gap by developing a time-aware deep learning species distribution model for the monarch butterfly (Danaus plexippus), an iconic species for biodiversity conservation. Using monarch occurrence records across the Americas gathered from scientific and citizen science sources, we embed the effect of monthly climatic variables in a sequential framework. We compare the performance of our seasonal model to conventional time-static baselines, showing not only better performance in the present, where the models have been trained and validated, but also in the past. Our findings show that climatic factors such as humidity, temperature, precipitation and cloud coverage strongly influence the ecological niche of the monarch butterfly, with notable seasonal and spatial variability. Applying our model under climate change scenarios, we predict a northwestward shift in the monarch range by the end of the XXI century, with expansion in Canada and significant contraction in California and Central America, key sites for overwintering that also host resident monarch populations. These changes could severely impact the species migratory cycle and population stability. Using Shapley values, an explainable AI technique, we identify the decrease in precipitation and humidity as important environmental drivers responsible for the contraction of overwintering sites. By focusing on a species of high ecological relevance through a time-aware modeling approach, this work brings novel insights for the conservation of migratory species in the face of the challenges posed by climate change.