  We finely describe the speed of "coming down from infinity" for birth and
death processes which eventually become extinct. Under general assumptions on
the birth and death rates, we firstly determine the behavior of the successive
hitting times of large integers. We put in light two different regimes
depending on whether the mean time for the process to go from $n+1$ to $n$ is
negligible or not compared to the mean time to reach $n$ from infinity. In the
first regime, the coming down from infinity is very fast and the convergence is
weak. In the second regime, the coming down from infinity is gradual and a law
of large numbers and a central limit theorem for the hitting times sequence
hold. By an inversion procedure, we deduce that the process is a.s. equivalent
to a non-increasing function when the time goes to zero. Our results are
illustrated by several examples including applications to population dynamics
and population genetics. The particular case where the death rate varies
regularly is studied in details.
