We consider a scattering process that takes place in a region of size R inside the static patch of the de Sitter spacetime such that R is smaller than the curvature length scale of the background. The effect of curvature can thus be studied perturbatively and we obtain the resultant corrections to the flat spacetime velocity memory effect.   

Then with Diksha Jain, we define a perturbative S-matrix in a local patch of de Sitter background in
the limit when the curvature length scale ($\ell$) is large and study the
'soft' behavior of the scalar QED amplitudes in de Sitter spacetime in generic
dimensions. We obtain the leading and subleading perturbative corrections to
flat space soft photon theorems in the large $\ell$ limit, and comment on the
universality of these corrections.