It is well-known that many diffusion equations can be recast as Wasserstein gradient flows. Moreover, in recent years, by modifying the Wasserstein distance appropriately, this technique has been transferred to further evolution equations and systems; see e.g. Maas (2011), Fathi and Simon (2016), Erbar (2016). In this paper we establish such a gradient flow representation for evolution equations that depend on a non-evolving parameter. These equations are connected to a local mean-field interacting spin system. We then use this gradient flow representation to prove a large deviation principle for the empirical process associated to this system. This is done by using the criterion established in Fathi (2016). Finally, the corresponding hydrodynamic limit is shown by using the approach initiated in Sandier and Serfaty (2004) and Serfaty (2011).