The increasing use of generative models such as diffusion models for synthetic data augmentation has greatly reduced the cost of data collection and labeling in downstream perception tasks. However, this new data source paradigm may introduce important security concerns. Publicly available generative models are often reused without verification, raising a fundamental question of their safety and trustworthiness. This work investigates backdoor propagation in such emerging generative data supply chain, namely, Data-Chain Backdoor (DCB). Specifically, we find that open-source diffusion models can become hidden carriers of backdoors. Their strong distribution-fitting ability causes them to memorize and reproduce backdoor triggers in generation, which are subsequently inherited by downstream models, resulting in severe security risks. This threat is particularly concerning under clean-label attack scenarios, as it remains effective while having negligible impact on the utility of the synthetic data. We study two attacker choices to obtain a backdoor-carried generator, training from scratch and fine-tuning. While naive fine-tuning leads to weak inheritance of the backdoor, we find that novel designs in the loss objectives and trigger processing can substantially improve the generator's ability to preserve trigger patterns, making fine-tuning a low-cost attack path. We evaluate the effectiveness of DCB under the standard augmentation protocol and further assess data-scarce settings. Across multiple trigger types, we observe that the trigger pattern can be consistently retained in the synthetic data with attack efficacy comparable to the conventional backdoor attack.