Large Language Models (LLMs) are widely applied to domain-specific tasks due to their massive general knowledge and remarkable inference capacities. Current studies on LLMs have shown immense potential in applying LLMs to model individual mobility prediction problems. However, most LLM-based mobility prediction models only train on specific datasets or use single well-designed prompts, leading to difficulty in adapting to different cities and users with diverse contexts. To fill these gaps, this paper proposes a unified fine-tuning framework to train a foundational open source LLM-based mobility prediction model. We conducted extensive experiments on six real-world mobility datasets to validate the proposed model. The results showed that the proposed model achieved the best performance in prediction accuracy and transferability over state-of-the-art models based on deep learning and LLMs.