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Online social network platforms have a problem with misinformation. One popular way of addressing this problem is via the use of machine learning based automated misinformation detection systems to classify if a post is misinformation. Instead of post hoc detection, we propose to predict if a user will engage with misinformation in advance and design an effective graph neural network classifier based on ego-graphs for this task. However, social networks are highly dynamic, reflecting continual changes in user behaviour, as well as the content being posted. This is problematic for machine learning models which are typically trained on a static training dataset, and can thus become outdated when the social network changes. Inspired by the success of continual learning on such problems, we propose an ego-graphs replay strategy in continual learning (EgoCL) using graph neural networks to effectively address this issue. We have evaluated the performance of our method on user engagement with misinformation on two Twitter datasets across nineteen misinformation and conspiracy topics. Our experimental results show that our approach EgoCL has better performance in terms of predictive accuracy and computational resources than the state of the art.