The fast-field program (FFP) has proven to be a useful tool for determining low-frequency sound fields generated by stationary sources in horizontally stratified atmospheres and oceans. Extension of the FFP to the case of moving sources is the topic of this paper. It is shown that the exact moving source solution involves a three-dimensional inverse Fourier transform (with respect to frequency and two horizontal wave numbers), that unfortunately is computationally infeasible. By shifting to source-following, Lorentz coordinates, however, the solution can be reduced to an approximate, one-dimensional, inverse transform involving only the horizontal, radial wave number. The result is an efficient computational method that works well at any subsonic Mach number, and includes accurate calculation of frequency Doppler effects in the presence of refraction. An example simulation of the noise field generated by a landing aircraft will be provided.