  The main subject of the paper is an escape from a multi-well metastable
potential on a time-scale of a formation of the quasi-equilibrium between the
wells. The main attention is devoted to such ranges of friction in which an
external saddle does not belong to a basin of attraction of an initial
attractor. A complete rigorous analysis of the problem for the most probable
escape path is presented and a corresponding escape rate is calculated with a
logarithmic accuracy. Unlike a conventional rate for a quasi-stationary flux,
the rate on shorter time-scales strongly depends on friction, moreover, it may
undergo oscillations in the underdamped range and a cutoff in the overdamped
range.
  A generalization of the results for inter-attractor transitions in stable
potentials with more than two wells is also presented as well as a splitting
procedure for a phenomenological description of inter-attractor transitions is
suggested.
  Applications to such problems as a dynamics of escape on time-scales shorter
than an optimal fluctuation duration, prehistory problem, optimal control of
fluctuations, fluctuational transport in ratchets, escapes at a periodic
driving and transitions in biased Josephson junctions and ionic channels are
briefly discussed.
