All Poses have a FoldTree definitionally. (A Pose contains a Conformation which contains a FoldTree and AtomTree). Ultimately AtomTree determines the kinematics (“how” stuff moves, or how internal coordinate changes propagate as 3-D coordinate changes.)
If you are moving the backbone, the standard method for controlling what residues can move is MoveMap.
If you are repacking or designing, the standard method of determining which sidechains can do what are PackerTask and TaskFactory.
MoveMap and PackerTask overlap in that both can make statements about the mobility of sidechains; the packer takes PackerTask and most other things take MoveMap.