We study theoretically the dynamics of a fluxon lattice (FL) in two coupled Josephson junctions. We show that when the velocity of the moving FL exceeds certain values (V_a,b), sharp resonances arise in the system which are related to the excitation of the optical and acoustic collective modes. In the interval (V_a, V_b) a reconstruction of the FL occurs. We also establish that one can excite localized nonlinear distortions (dislocations) which may propagate through the FL and carry an arbitrary magnetic flux.