In this work, we introduce a novel motion-robust spatially constrained parameter estimation (MOSCOPE) technique for kidney diffusion-weighted MRI. The proposed motion compensation technique does not require a navigator, trigger, or breath-hold but only uses the intrinsic features of the acquired data to track and compensate for motion to reconstruct precise models of the renal diffusion signal. We have developed a technique for physiological motion tracking based on robust state estimation and sequential registration of diffusion sensitized slices acquired within 200ms. This allows a sampling rate of 5Hz for state estimation in motion tracking that is sufficiently faster than both respiratory and cardiac motion rates in children and adults, which range between 0.8 to 0.2Hz, and 2.5 to 1Hz, respectively. We then apply the estimated motion parameters to data from each slice and use motion-compensated data for 1) robust intra-voxel incoherent motion (IVIM) model estimation in the kidney using a spatially constrained model fitting approach, and 2) robust weighted least squares estimation of the diffusion tensor model. Experimental results, including precision of IVIM model parameters using bootstrap-sampling and whole kidney tractography, showed significant improvement in precision and accuracy of these models using the proposed method compared to models based on the original data and volumetric registration.