Study the docs under DYNAMICS > GOALS
I would either create and key the goalPP influence on each vertice goal weight, OR create a simple animated ramp to the goal object to drive the goal weights so your particles are attracted to the object where and how you want
Hope that helped! Good luck!
PS - Here is a blurb from the docs:
When you add a goal to an object, Maya adds a corresponding goal weight attribute to the trailing particle object. The goal weight sets how much all particles of the trailing object are attracted to the goal. You can set the goal weight before you create the goal using the Goal options window, or you can adjust the goal weight afterwards using the Attribute Editor.
You can set goal weight to a value between 0 and 1. A value of 0 means that the goal's position has no effect on the trailing particles. A value of 1 moves the trailing particles to the goal object position immediately.
A value between 0 and 1 causes the particles to move toward the goal as if bound to it by an elastic spring. The closer to 1, the faster the trailing particles settle at their final position. The goal weight value is 0.5 by default.
To make the trailing particle object follow the goal object with less oscillation, set the trailing particle object's Conserve attribute to a value less than 1, for instance, 0.8. A value of 0 prevents oscillation.
You can turn the effect of a goal object off or on without setting goal weight. With the trailing particle object attributes displayed in the Attribute Editor, turn off the Goal Active attribute associated with the goal object.
Israel "Izzy" Long
Motion and Title Design for Broadcast-Film-DS