Turn off an effect on certain frame

Keyframing the time offset of echoes will make the copies roll under each other.  If the time offset is zero when you want the echoes gone, then they will all be right under the original layer, and you can set the number of echoes to zero. This will start with the echoes strung out behind the movement and then move the whole string toward the original circle.

The other option is to animate the number of echos, but they will snap off as the numbers decrease because there are no half copies. 

You can also split the layer, turn off Echo, and animate the opacity of the layer with the effect to remove the echoes. It all depends on how you want the motion to look. I would probably go with animating the offset so the echoes quickly stacked up under the moving circle when it stopped for the first time, but I would probably keep a little offset during the bounce to maintain the illusion of speed.

You could animate the number of echos down to zero at the time when you don't want the effect any more. You could do it with keyframes or with an expression tied to your other animation keyframes if you can define the timing relationship you want.