  We study the asymptotic hitting time $\tau^{(n)}$ of a family of Markov
processes $X^{(n)}$ to a target set $G^{(n)}$ when the process starts from a
trap defined by very general properties.
  We give an explicit description of the law of $X^{(n)}$ conditioned to stay
within the trap, and from this we deduce the exponential distribution of
$\tau^{(n)}$. Our approach is very broad ---it does not require reversibility,
the target $G$ does not need to be a rare event, and the traps and the limit on
$n$ can be of very general nature--- and leads to explicit bounds on the
deviations of $\tau^{(n)}$ from exponentially. We provide two non trivial
examples to which our techniques directly apply.
