Recent advances in laser wakefield acceleration demonstrated the generation of extremely short (with a duration of a few femtoseconds) relativistic electron bunches with relatively low (of the order of couple of percent) energy spread. In this article we study the dynamics of such bunches in drift space (vacuum) and in channel-guided laser wakefields. Analytical solutions were found for the transverse coordinate of an electron and for the bunch envelope in the wakefield in the case of arbitrary change in the energy. Our results show strong bunch dynamics already on a millimeter scale propagation distance both in plasma and in vacuum. When the bunch propagates in vacuum, its transverse sizes grow considerably; the same is observed for the normalized bunch emittance that worsens the focusability of the bunch. A scheme of two-stage laser wakefield accelerator with small drift space between the stages is proposed. It is found that fast longitudinal betatron phase mixing occurs in a femtosecond bunch when it propagates along the wakefield axis. When bunch propagates off axis, strong bunch decoherence and fast emittance degradation due to the finite bunch length was observed.
