An investigation is made of the dynamics of fluxon chains in long Josephson junctions with a periodic lattice of local inhomogeneities. In the commensurate case a chain as a whole is in a pinned state as long as the density of the bias current density is below a certain critical value. It is shown that defects in the form of an excess fluxon or a "hole" may propagate in a pinned chain. The long-wavelength approximation is used to deduce the evolution equation of a local deformation of a chain: the result is an "elliptic sine-Gordon equation" which has exact soliton solutions ("supersolitons") describing such defects. The current-voltage characteristics are found for the motion of a supersoliton in the presence of dissipation and a bias current (when the density of this current is less than the critical value). Supersoliton excitations are then predicted on the basis of a direct numerical solution of a perturbed sine-Gordon equation describing a periodically inhomogeneous junction. The soliton solutions of the elliptic sine-Gordon equation are also obtained numerically. Although the latter equation is in all probability nonintegrable, a numerical investigation shows in particular that a collision of two solitons of opposite polarities is in practice absolutely elastic. ... mehrBoth models are used to calculate the current-voltage characteristics of a ring-shaped inhomogeneous junction. An experimental study is reported of a linear Josephson junction containing a regular lattice of deliberately formed inhomogeneities. Steps on the current-voltage characteristics of such a junction are found to occur at a voltage that depends strongly on the applied magnetic field. These features are attributed to the motion of supersolitons in a junction.