The character should have a class Locomotor that takes care of its movement along the path. That way, it can walk/drive/crawl/swim depending on its capabilities and state.

Let's say I have AI that wants to go from one point to another. It finds path and starts to move. What do I do here? I just move character along path? Or use some sort of character controller (line the one I use for playable character to move)?

Depends on the game. I have a third person shooter game with bots, the AI basically acts just like a player, and only gives inputs just like a player would. So the AI would find a path, then translate that path into the current input to report.

There are as many possible answers to this as there are game implementations in the universe.

Do what makes sense for your code base.