Video editing software that will follow a subject in a cropped area of the video
You have two options:


Keyframes:
Most a bit more advanced video editing programs have key frames, with them you say at time X, crop to Position Y. The computer will automatically fill the motion.
Depending on the scene, this method needs a lot of work tho.
Tracking:
Another way is to track the subject. That's about what you said in the question. You specify an area and the computer follows it. A freeware with this feature would be DAVINCI RESOLVE or Hitfilm. But both software require some basic knowledge of video editing.