Some parts of my characters refuses to move.

Have you checked your character rollbones?
When roll bones are not inline with the arm chain hierarchy or leg hierarchy, and thus are not animated when the arm is animated. When there is skinning on these bones, the vertices skinned to the roll bones appears as not moving while the mesh skinned to the rest of the arm chain is moving.
Actually, if you try to animate the limb directly from the fbx file, you will see the same kind of problems.
There might be some controllers in Maya that allows to control the position of those roll bones when the rest of the limb is moving, but  Golaem Crowd representation doesn't include these controllers.
For now, there are 2 options for your character to work with Golaem Crowd:
- reskinning the character to ensure nothing is skinned on the roll bones;
- changing the hierarchy of bones to have roll bones inline with the arm bone chain and re-create the .gcha from there.