In the laser wakefield accelerator (LWFA) a short intense laser pulse, with a duration of the order of a plasma wave period, excites an unusually strong plasma wake wave (laser wakefield). Recent experiments on laser wakefield acceleration [Nature (London) 431, p.535, p.538, p.541 (2004)] demonstrated generation of ultra-short (with a duration of a few femtoseconds) relativistic electron bunches with relatively low energy spread of the order of a few percent. We have studied the dynamics of such bunches in vacuum and in laser wakefield. The results show strong bunch dynamics already on a few millimeters propagation distance in both cases. In vacuum, the bunch radius and emittance quickly grow. The latter worsens the focusability of the bunch. We found that when a femtosecond bunch is accelerated in a channel-guided laser wakefield, for realistic bunch lengths, the bunch length is approximately conserved. However, the spread in betatron frequencies leads to fast betatron phase mixing in the bunch envelope for on-axis injection. When bunch is injected in a laser wakefield off-axis, the bunch decoherence results in considerable increase in the normalized bunch emittance, and, in some cases, in increase in the energy spread, after acceleration. We also discuss a possible two-stage laser wakefield accelerator.