Great question - there are SO MANY WAYS to achieve the effect you are talking about. A few of my favorites (at the moment) are:
- Doppler plugins: Waves Doppler and GRM Doppler are the most common and yield instant results with very little effort. Unfortunately, these results can often sound pretty generic and lifeless, depending on your source material.
- Level and EQ manipulation: This method is a throwback to the way "they used to do it". In short, by riding the fader level (amplitude) and tweaking EQ dips and curves, you can approximate the Doppler effect as it would sound in the real world. Because, if you think it through, all the Doppler effect is is "the change in frequency of a wave for an observer moving relative to its source". As a sound moves towards or away from you, not only is the amplitude changing but also the way the sound is moving through the air. Riding levels and twisting EQ knobs will deliver that same effect.
- Flanging: This one is easily overlooked. A very slight amount of flanging can trick the brain into thinking it hears the effects of motion on a sound.
- Stand-alone software: Programs like MetaSynth, NI Whoosh Machine and WaveWarper employ a bunch of processing/automating to achieve this effect. I've especially enjoying the WaveWarper method lately as you can manually drive layers of sounds over the Doppler point, giving you control over when the sound peaks and how the EQ will affect that peak point.
Just a few ideas, your mileage may vary...