The kinetics of single transition A-->B over a fluctuating barrier is considered. Fluctuations are modeled by dichotomous noise. The average first passage time (AFPT) t(p)* is defined as the time elapsed from the beginning of the process (system in the state A with probability 1) to the moment when the system attains for the first time the state B with the average probability equal to p. It is found that the non-Markovianity of the barrier fluctuations may introduce oscillations in the process of barrier crossing and in the effective reaction rate, and elongate the AFPTs. Especially, t(0.95)* may become infinite, even when t(0.5)* remains finite-the process of barrier crossing is reverted after some time. However, in some cases (strong asymmetry of barrier fluctuations, high AFPTs in the absence of fluctuations, together with long memory characteristic time of the non-Markovian part of the fluctuations) the effective reaction rates can be enhanced and AFPTs shortened in comparison with those for static barrier. (C) 1997 American Institute of Physics.