It has been demonstrated theoretically and experimentally that an estimate of the Green's function between two receivers can be obtained by cross-correlating acoustic (or elastic) ambient noise recorded at these two receivers. Coherent wavefronts emerge from the noise cross-correlation time function due to the accumulated contributions over time from noise sources whose propagation path pass through both receivers. Previous theoretical studies of the performance of this passive imaging technique have assumed that no relative motion between noise sources and receivers occurs. In this article, the influence of noise sources motion (e.g., aircraft or ship) on this passive imaging technique was investigated theoretically in free space, using a stationary phase approximation, for stationary receivers. The theoretical results were extended to more complex environments, in the high-frequency regime, using first-order expansions of the Green's function. Although sources motion typically degrades the performance of wideband coherent processing schemes, such as time-delay beamforming, it was found that the Green's function estimated from ambient noise cross-correlations are not expected to be significantly affected by the Doppler effect, even for supersonic sources. Numerical Monte-Carlo simulations were conducted to confirm these theoretical predictions for both cases of subsonic and supersonic moving sources.