6G networks are envisioned to support on-demand AI model downloading to accommodate diverse inference requirements of end users. By proactively caching models at edge nodes, users can retrieve the requested models with low latency for on-device AI inference. However, the substantial size of contemporary AI models poses significant challenges for edge caching under limited storage capacity, as well as for the concurrent delivery of heterogeneous models over wireless channels. To address these challenges, we propose a fine-grained AI model caching and downloading system that exploits parameter reusability, stemming from the common practice of fine-tuning task-specific models from a shared pre-trained model with frozen parameters. This system selectively caches model parameter blocks (PBs) at edge nodes, eliminating redundant storage of reusable parameters across different cached models. Additionally, it incorporates coordinated multipoint (CoMP) broadcasting to simultaneously deliver reusable PBs to multiple users, thereby enhancing downlink spectrum utilization. Under this arrangement, we formulate a model downloading delay minimization problem to jointly optimize PB caching, migration (among edge nodes), and broadcasting beamforming. To tackle this intractable problem, we develop a distributed multi-agent learning framework that enables edge nodes to explicitly learn mutual influence among their actions, thereby facilitating cooperation. Furthermore, a data augmentation approach is proposed to adaptively generate synthetic training samples through a predictive model, boosting sample efficiency and accelerating policy learning. Both theoretical analysis and simulation experiments validate the superior convergence performance of the proposed learning framework.