The demo character controller (designed just for demonstrating the IK capabilities, not to be some sort of universal character controller) just doesn't have any strafe running animations, so probably its not moving because you have "Walk By Default" enabled in UserControlThirdPerson. You can add running animations to the strafing blend tree if you have some. Kubold's Movement Animset Pro would probably work nicely.
The demo controller doesn't have any move speed modifier, you could add a float parameter to the controller (make a duplicate of it so your work would not be overwritten the next time you update FIK). Use that param as soeed multiplier for the "Grounded Strafe" animation state.