A moving diffractor changes the energy of elastically diffracted neutrons by the Doppler effect. Depth-graded multilayers can diffract neutrons over a large band of energy. Using a pulsed neutron source, such a depth-graded multilayer, decelerating synchronously with the incident neutron pulse, can shift the reflected neutrons into a compressed energy window. This focusing in energy is associated with a broadening of the pulse in time, but the process does not involve a significant decrease in the neutron phase-space density. The proposed method can be used to design long pulse or quasi-continuous sources of cold, very cold or ultra cold neutrons (UCN). The analysis concentrates on enhanced production of UCN at pulsed neutron sources.