Multiple and supersonic topological excitations (kinks) driven by an external dc force in the Frenkel-Kontorova model (a chain of atoms subjected to a periodic substrate potential) with the exponential interatomic interaction are studied with the help of numerical simulation. The simulation results are interpreted in terms of dynamics of two limiting cases, the exactly integrable sine-Gordon equation and the Toda chain. The stability of driven kinks and scenarios of their destruction are described for a wide range of model parameters.