Detect movement on moving robot? PIR Sensor
Basically you have a robot and sensors that can detect movement.
There is one caveat though. For the PIR sensors to (non false-positive) detect motion, they have to be stationary (as far as I know).

So with only these PIR sensors it's unlikely that you can track something well.
Tracking something would be done best by using a range/proximity sensor.



The image below, seems to be a cute robot, with eyes.
You could take 3 sensors and place them under an angle (one forward, one forward/left and one forward/right) [picture 2]. Or just have one that rotates (will be more difficult) [picture 1].

You would want the robot to start following anything that comes close (or when one of the sensors sees a difference in distance. And then the robot will try to keep that distance, brake when he gets closer, drive when he gets further.

If that's working, you can implement the sensors at the side, that if suddenly he loses track of the object/person, he'll adjust his alignment so that the middle sensor sees it again. This could also be done by simpler sensors [picture 3]

When the robot gets to close, and the other object doesn't try to run for quite a time, you can make him check for another target. (With a PIR or by rotating and stopping rotating to check range/pir)

I believe this solution is more applicable as with only PIR sensors, as it's usefull to know the range when you're following something. You don't want to bash into it.