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Known examples of the holographic dictionary in asymptotically Anti-de Sitter spacetimes equate moduli spaces of bulk vacua with conformal manifolds in the dual quantum field theory. We demonstrate that the same identification holds for gravity in asymptotically flat spacetimes in any dimension, in accord with expectations derived from the celestial conformal field theory (CCFT) formalism. Soft limits of moduli scalars described by the sigma model are universal, and relate to parallel transport of $S$-matrix observables over the moduli space of bulk vacua. The leading "soft moduli operator" is the shadow transform of a dimension $Δ=d$ marginal operator $M(x)$. The universal form of the soft limit guarantees that $M(x)$ acts as a marginal deformation in the CCFT$_d$, and coherent states of the soft scalars correspond to finite deformations along the conformal manifold. This manifold typically has curvature, which is captured by the antisymmetric double-soft theorem and which reflects the Berry curvature in CCFT$_d$. We also compute the Mellin-transformed four-point function in the sigma model and compare to a formula of Kutasov for the curvature of the conformal manifold.