If your character is not moving, but its playing the animations at it supposed to do it, Usually this is because the character is not using Root Motion Animations, but InPlace Animations.
Root Motion
The animation moves the character .
In Place
The animation is done "In Place", there's no character movement
Steps
Go to the Speed Sets tabs and select the Ground Speed.
On the POSITION parameter of any Speed Modifier(Walk,Trot,Run), add a value higher than 0.
That will make the animal move without RootMotion.
Try Different Lerp Values to modify the Responsiveness.
Do the Same for the Rotation parameter... to increase the Turning Speed.
