The correlation between two pulse-echo ultrasound signals is used to achieve a wide range of ultrasound techniques, such as Doppler imaging and elastography. Prior theoretical descriptions of pulse-echo correlations were restricted to stationary scatterers. Here, a theory for the correlation of moving scatterers is presented. An expression is derived for the correlation of two pulse-echo signals with arbitrary transmit and receive apertures acquired from a medium undergoing bulk motion using the Fresnel approximation. The derivation is shown to coincide with prior derivations in the absence of scatterer motion. The theory was compared against simulations in applications of phase-shift estimation and aperture coherence measurements. The phase-shift estimate and jitter were accurately predicted under axial and transverse motion for focused transmit apertures and for sequential and interleaved synthetic transmit apertures. The theory also accurately predicted how motion affects the correlation coefficient between receive aperture elements for a synthetic transmit aperture. The presented theory provides a framework for analyzing the correlations of arbitrary pulse-echo configurations for applications in which scatterer motion is expected.