3d cursor x,y,z does not move with mesh I have a model that has a root (main) bone added to rear of pelvis. The root bone
is for movement (animation) of model. 
 When I'm in pose mode and move model forward using the root bone, I noticed the   x,y,z 3d cursor of the mesh does not move with the mesh. 

I have a feeling this will cause problems if I continue to animate. I've decided to stop researching and ask for help. I have posted 3 pictures. The first picture shows model before movement, second picture is after I move Model. The third picture is what it looks like in object mode.

Is this normal? Do I need the 3d cursor xyz  to stay with mesh when I move model in pose mode? If so how.
This behaviour is absolutely fine: the bones move vertices only, which have some weight related to their respective vertex group (with the same name). The object (its origin) stays still. If you move the object in object mode you will have weird issues unless you move the rig accordingly.

The usual way to animate is setting both object and rig at the center of the world in object mode, then in pose mode use the root bone to grab your character where he needs to be, and start the animation from there. The root bone can stay still while the rest of the bones are animating the mesh, or can be moved along a path in case of walk cycles or similar.