Howdy,
Excellent question! Accomplishing this is pretty easy with the Motion Path behavior. This applies, by the way, to any positional-animation scenario, not just your particle emitter. In brief: you draw a shape (in Motion) that will indicate your animation path, you apply the Motion Path behavior to your emitter, and you tell the behavior to use the shape for the animation. So…
1. Use the shape tool (B) to draw your desired animation path (bezier, b-splines, whatever).
2. Apply the Motion Path behavior to your emitter.
3. With the Motion Path behavior selected, go to the Inspector and view the Behaviors tab. Set the Path Shape popup to ‘Geometry’.
4. Drag-and-drop the shape you drew in Step 1 into the Shape Source image-well that is now visible in the Inspector (or you can pick any shape object in the project from the little ‘To’ popup menu that is next to the image-well).
Now your emitter will move along the shape’s path over the duration of the behavior. Want to change the path? Go ahead and edit the shape. Want it to be faster or slower? Change the behavior’s duration. You can also change the behavior’s Speed popup to get Ease In/Out, and such. No keyframes required.
Bonus Round
This last bit is a power-user tip of the sort that we used to pay Brian Maffitt $600 for on the Total Training AE tapes: the ‘Custom’ option in the Motion Path behavior’s Speed popup menu. When you pick ‘Custom’, a new Custom Speed parameter is revealed, which goes from 0-100% and has two default keyframes on it, at the beginning and end of the behavior. Think of this as the “completion” value of the behavior on any given frame. So the default is that it animates from 0 to 100% over the behavior’s duration, but you can view this curve in the Keyframe Editor, giving you sick control over the animation’s progress, including stuff like reversals, etc. So if you want the behavior to cruise along for 60 frames and then hold for a bit, then reverse its progress, then surge ahead, you can keyframe that with one simple speed curve. Then you tell your client how you had to move the Earth and the Moon to do it all in so little time 🙂