The Join-the-Shortest-Queue-d routing policy is considered for a large system with $n$ servers. Moderate deviation principles (MDP) for the occupancy process and the empirical queue length process are established as $n\to \infty$. Each MDP is formulated in terms of a large deviation principle with an appropriate speed function in a suitable infinite-dimensional path space. Proofs rely on certain variational representations for exponential functionals of Poisson random measures. As a case study, the convergence of rate functions for systems with finite buffer size $K$ to the rate function without buffer is analyzed, as $K \to \infty$.