We consider the tree-level scattering of massless particles in $(d+2)$-dimensional asymptotically flat spacetimes. The $S$-matrix elements are recast as correlation functions of local operators living on a space-like cut $Md$ of the null momentum cone. The Lorentz group $SO(d+1,1)$ is nonlinearly realized as the Euclidean conformal group on $Md$. Operators of non-trivial spin arise from massless particles transforming in non-trivial representations of the little group $SO(d)$, and distinguished operators arise from the soft-insertions of gauge bosons and gravitons. The leading soft-photon operator is the shadow transform of a conserved spin-one primary operator $Ja$, and the subleading soft-graviton operator is the shadow transform of a conserved spin-two symmetric traceless primary operator $Tab$. The universal form of the soft-limits ensures that $Ja$ and $Tab$ obey the Ward identities expected of a conserved current and energy momentum tensor in a Euclidean CFT, respectively.