Dynamics of fluxons in a discrete Josephson transmission line is investigated, combining numerical simulations and an analytical approach. It is found that, in different ranges of the parameters (the driving dc bias current and dissipative constant), two fluxons (2π−kinks) may form either a bifluxon
(4π−kink), or various bound states (2π+2π−kinks with a finite separation), which can stably propagate along the line. The stability of these states is investigated as a function of the kink velocity. An analytical approach is based on prediction of formation of a two-kink bound state through the interaction mediated by their oscillating “tails.” At small velocities, a satisfactory agreement is found between the analysis and the numerical results. At still smaller velocities, a new phenomenon is predicted analytically and found numerically, viz., transition from an asymmetric “tailed” kink to a symmetric tailless one. Conditions for experimental observation of the predicted behavior, as well as its practical consequences for the fluxon propagation in the discrete Josephson transmission lines, are discussed too.