  A general theory is derived for the moments of the first passage time of a
one-dimensional Markov process in presence of a weak time-dependent forcing.
The linear corrections to the moments can be expressed by quadratures of the
potential and of the time-dependent probability density of the unperturbed
system or equivalently by its Laplace transform. If none of the latter
functions is known, the derived formulas may still be useful for specific cases
including a slow driving or a driving with power at only small or large times.
In the second part of the paper, explicite expressions for mean and variance of
the first passage time are derived for the cases of a linear or a parabolic
potential and an exponentially decaying driving force. The analytical results
are found to be in excellent agreement with computer simulations of the
respective first-passage processes. The particular examples furthermore
demonstrate that already the effect of a simple exponential driving can be
fairly involved implying a nontrivial nonmonotonous behavior of mean and
variance as functions of the driving's time scale.
