The OBJ file contains an address where it can find the MTL directory. When you move the OBJ file, you should also move the MTL file.
1 - If OBJ and MTL are in the same directory, they should stay in the same directory.
2 - If the OBJ is in [font=courier new]parent[/font] directory and MTL is in [font=courier new]parent/textures[/font] directory, and you move the OBJ file to [font=courier new]model[/font] directory, then the MTL should move to [font=courier new]model/textures[/font] directory.