Optical coherence tomography (OCT)-based elasticity imaging can map soft tissue elasticity based on speckle-tracking of elastic wave propagation using highly sensitive phase measurements of OCT signals. Using a fixed elastic wave source and moving detection, current imaging sequences have difficulty in reconstructing tissue elasticity within speckle-free regions, for example, within the crystalline lens of the eye. We present a moving acoustic radiation force imaging sequence to reconstruct elastic properties within a speckle-free region by tracking elastic wave propagation from multiple laterally moving sources across the field of view. We demonstrate the proposed strategy using heterogeneous and partial speckle-free tissue-mimicking phantoms. Harder inclusions within the speckle-free region can be detected, and the contrast-to-noise ratio slightly enhanced compared to current OCE imaging sequences. The results suggest that a moving source approach may be appropriate for OCE studies within the large speckle-free regions of the crystalline lens.