Magnetic resonance (MR) imaging, including cardiac MR, is prone to domain shift due to variations in imaging devices and acquisition protocols. This challenge limits the deployment of trained AI models in real-world scenarios, where performance degrades on unseen domains. Traditional solutions involve increasing the size of the dataset through ad-hoc image augmentation or additional online training/transfer learning, which have several limitations. Synthetic data offers a promising alternative, but anatomical/structural consistency constraints limit the effectiveness of generative models in creating image-label pairs. To address this, we propose a diffusion model (DM) trained on a source domain that generates synthetic cardiac MR images that resemble a given reference. The synthetic data maintains spatial and structural fidelity, ensuring similarity to the source domain and compatibility with the segmentation mask. We assess the utility of our generative approach in multi-centre cardiac MR segmentation, using the 2D nnU-Net, 3D nnU-Net and vanilla U-Net segmentation networks. We explore domain generalisation, where, domain-invariant segmentation models are trained on synthetic source domain data, and domain adaptation, where, we shift target domain data towards the source domain using the DM. Both strategies significantly improved segmentation performance on data from an unseen target domain, in terms of surface-based metrics (Welch's t-test, p < 0.01), compared to training segmentation models on real data alone. The proposed method ameliorates the need for transfer learning or online training to address domain shift challenges in cardiac MR image analysis, especially useful in data-scarce settings.