``Do Carroll particles move?'' The answer depends on the characteristics of the particle such as its mass, spin, electric charge, and magnetic moment. A massive Carroll particle (closely related to fractons) does not move; its immobility follows from the Carroll boost symmetry. A strictly massless Carroll particle may propagate by following the Hall law consistently with partial breaking of their Carroll boost symmetry. In $d=2$ space dimensions, the Carroll group has a two-fold central extension which allows us to generalize the dynamics to massive and massless particles, including anyons. The anyonic spin and magnetic moment combine with the doubly-extended structure parameterized by two Casimir invariants interpreted as intrinsic magnetization and non-commutativity parameter. The extended Carroll particle subjected to an electromagnetic background field moves following a generalized (anomalous) Hall law which includes a Zeeman force. Our theory is illustrated by massless, uncharged anyons with doubly-centrally extended structure (we call ``exotic photons'') which move on the horizon of a Kerr-Newman Black Hole, giving rise to an anyonic spin-Hall Effect.