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Product configuration can be defined as the task of tailoring a product according to the specific needs of a customer. Due to the inherent complexity of this task, which for example includes the consideration of complex constraints or the automatic completion of partial configurations, various Artificial Intelligence techniques have been explored in the last decades to tackle such configuration problems. Most of the existing approaches adopt a single-site, centralized approach. In modern supply chain settings, however, the components of a customizable product may themselves be configurable, thus requiring a multisite, distributed approach. In this paper, we analyze the challenges of modeling and solving such distributed configuration problems and propose an approach based on Distributed Constraint Satisfaction. In particular, we advocate the use of Generative Constraint Satisfaction for knowledge modeling and show in an experimental evaluation that the use of generic constraints is particularly advantageous also in the distributed problem solving phase.