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This paper introduces a novel pipeline for summarising timelines of events reported by multiple news sources. Transformer-based models for abstractive summarisation generate coherent and concise summaries of long documents but can fail to outperform established extractive methods on specialised tasks such as timeline summarisation (TLS). While extractive summaries are more faithful to their sources, they may be less readable and contain redundant or unnecessary information. This paper proposes a preference-based reinforcement learning (PBRL) method for adapting pretrained abstractive summarisers to TLS, which can overcome the drawbacks of extractive timeline summaries. We define a compound reward function that learns from keywords of interest and pairwise preference labels, which we use to fine-tune a pretrained abstractive summariser via offline reinforcement learning. We carry out both automated and human evaluation on three datasets, finding that our method outperforms a comparable extractive TLS method on two of the three benchmark datasets, and participants prefer our method's summaries to those of both the extractive TLS method and the pretrained abstractive model. The method does not require expensive reference summaries and needs only a small number of preferences to align the generated summaries with human preferences.