Hey @Ivanim, thanks for the screenshots!
Something could have went wrong went exporting/importing the jacket mesh. You can see in the second image that the spine bones are not highlighted, which means they do not exist in that file or do not have any influence on the mesh.
If you look at your character, or any of the other meshes like the head, feet, legs, etc, they will probably have the bones highlighted like this, meaning they have influence on the vertices of the mesh:
If you move the spine bones in the jacket preview window, it will probably not move, meaning it has all the influence assigned to the root bone or to another set of joints that are not being controlled by the character animation.
If the weights are correctly assigned to the spine joints in blender, maybe you can try exporting with different settings. If not, you might want to try copying the weights of the main body into the jacket in blender before re-exporting.
I hope that helps!