The influence of a cosφ conductance on the motion of fluxons in long and narrow Josephson junctions is investigated by numerical computations and by a perturbation analysis. It turns out that the presence of the cosφ term will have opposite effects on the motion of a fluxon and on plasma waves or breathers. If the fluxon motion is damped, the plasma waves are enhanced by the cosφ term and vice versa. The presence of loss and bias results, in any event, in stabilization of the fluxon in a stationary motion. Good agreement between the numerical result for fluxon motion and the perturbation analysis is found.