Paul,
In FCP it is harder to do than say in Motion or After Effects, but it can be done. Without keyframing this manually you’d need some kind of tracker, I believe CHV makes a pretty good one.
Joes filters have a plug to blur or pixilate these things, but then you need to animate (track) it to the shot.
Or you can just duplicate the shot over itself, and use a garbage matt to isolate the plate, add blur or whatever but you’d still have to keyframe the size and position of the matte through the shot.
Have a wonderful day.