The delivery of continuous and seamless Connected and Automated Mobility (CAM) services across international borders poses substantial challenges due to the complex interactions among 5G mobile network operators (MNO), road and rail infrastructure operators, neutral host operators, and end users, at both sides of the border. Some of the most prominent challenges include achieving roaming with short-enough interruption time and low latency, and providing cross-MNO orchestration and network slicing continuity. The EU-funded Horizon 2020 5GMED project has devised a cross-border network architecture that directly addresses such challenges in smart mobility use cases across the Spanish-French border. This paper presents an overview of the 5GMED use cases, technical challenges, and ongoing research and development efforts. It provides implementation details and lessons learnt from the project's small-scale and large-scale trials that revolve around four use cases: remote driving, road infrastructure digitization, future railway mobile communications, and follow-me infotainment.