Doppler direction finders work better on a steady CW source, but they usually also work fine on "reasonably" modulated signals. The principle of operation is normally not exactly what you wrote; in other words, nobody actually wiggles an antenna physically -- it is typically done electronically by switching on one antenna at a time in a horizontal array spaced 1/4 wave apart. That results in a tone superimposed on the received signal resulting from the simulated Doppler effect of "moving" and antenna, and the bearing to the signal source is related to the phase of the tone relative to the switching tone as a reference.

The only time I have trouble using my Ramsey Electronics DDF-1 is when the signal has lots of spectral spikes drifting through the audio passband of my FM receiver, e.g. when I was hunting for the source of interference in an apartment complex where someone had a sick router box leaking out RF in and around the 2m band.

Andy KR6DD