Synthetic aperture radar (SAR) provides high resolution images that are well suited for through-the-wall target detection and recognition. As targets behind-the-wall undergo non-uniform motions, such as vibration, rotation and acceleration, their patterns can be recognized. To understand these signatures in through-the-wall SAR, we model and analyze the non-uniform motion-induced Doppler effect as well as the focused target SAR image. In particular, the wall effects on the focused SAR image and the micro-Doppler are formulated and analyzed. These analyses facilitate improving the target recognition performance by quantitatively estimating the parameters of the micro-Doppler signatures as well as the SAR imaging. We further analyze the detection performance of the non-uniform motion-induced target based on the generalized likelihood ratio test (GLRT) technique. The relationship between motion parameters and the detection performance allows us to evaluate the performance bound and the minimum detectable parameters.