Which (Fourier transform?) processing method to use for time variant audio processing? (newbie)
I believe the answer to your question is Short-time Fourier Transform or Short-term Fourier Transform.  There is the wikipedia article.  I tried to show the essential math in this answer.

Both the Phase-Vocoder and Sinusoidal Modeling are done with the STFT of some form or another.  The two methods sorta merge in concept at the STFT level.  Remember that the analysis window for STFT need not be the same as the synthesis window in constructing the processed audio for output.