Most astrophysical objects are moving due to an interaction with their environment or due to the motion of their host system. Gravitational wave sources are no exception. In contrast, most gravitational wave sources are likely moving with velocities of 100s to 1000s km/s since they usually form in the densest regions of galaxies. We show that when considering a complete description of gravitational waves, and not only their frequency, a constant center-of-mass velocity leads to a change of the wave similar to the 'beaming effect' of electromagnetic waves. This alteration of the wave then in turn leads to a time-dependent phase shift which together with the excitation of the spherical modes of the wave can be used to detect the motion of the source from gravitational waves alone. Thus not only breaking the well-known mass-redshift degeneracy of gravitational waves but also allowing us to study the dynamics of the source's environment and host system.