The dynamics of fronts, or kinks, in dispersive media with gain and losses is considered. It is shown that the front parameters, such as the velocity and width, depend on initial conditions. This result is not typical for dissipative systems. For exponentially decreasing initial conditions, the relations for the front parameters are found. A presence of the global bifurcation, when a soliton solution is replaced by the front solution, is demonstrated. It is also shown that in order to observe fronts, the front velocity should be larger than the characteristic velocity of the modulational instability.