Rigid Dynamics uses rigid bodies by default, so the deformation in the links is always zero. Once the motion loads are transferred to Static Structural, then the link is flexible and there is a nonzero deformation.
It is possible that the boundary conditions and initial conditions in the Transient Dynamics model are not identical to the Rigid Dynamics model.
Even if the conditions are identical, the flexibility in all parts can change the forces going through the link of interest.
I doubt that different timesteps between the dynamic models is responsible for differences at a specific time, but you could also ensure that the timesteps are equal.
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