Large-scale diffusion models are adept at generating high-fidelity images and facilitating image editing and interpolation. However, they have limitations when generating images in evolving domains. This paper introduces Terra, a novel Time-varying low-rank adapter that offers a fine-tuning framework for domain flow generation. The key design in Terra involves constructing a continuous parameter manifold via a time variable with its expressive power theoretically analyzed. Furthermore, this framework offers a generation-based approach to address the domain shift problems in unsupervised domain adaptation and domain generalization. Specifically, Terra transforms images from the source domain to the target domain and generates interpolated domains with various styles to bridge the gap between domains and enhance the model generalization, respectively. Extensive experiments on various benchmark datasets empirically demonstrate the effectiveness of Terra.