Frequency-modulated (FM) frequency combs constitute an exciting alternative to generate equidistant spectra. The full set of Maxwell-Bloch equations is reduced to a single master equation for lasers with fast gain dynamics to provide insight into the governing mechanisms behind phase locking. It reveals that the recently observed linear frequency chirp is caused by the combined effects of spatial hole burning, group velocity dispersion, and Kerr nonlinearity due to asymmetric gain. The comparison to observations in various semiconductor lasers suggests that the linear chirp is general to self-starting FM combs.