  We present a dynamical theory for the phase differences along a stacked
direction of intrinsic Josephson junctions (IJJ's) in layered multigap
superconductors, motivated by the discovery of highly-anisotropic iron-based
superconductors with thick perovskite-type blocking layers. The dynamical
equations describing AC and DC intrinsic Josephson effects peculiar to multigap
IJJ's are derived, and collective Leggett mode excitations in addition to the
Josephson plasma established in single-gap IJJ's are predicted. The dispersion
relations of their collective modes are explicitly displayed, and the
remarkable peculiarity of the Leggett mode is demonstrated.
