  We give indications that outer future trapping horizons play a role in the
particular semi-classical instability of an evolving black hole that produces
the Hawking's radiation. These are obtained with the use of the Hamilton-Jacobi
tunneling method. It automatically selects one special expression for the
surface gravity of a changing horizon, the one defined a decade ago by Hayward
using Kodama's theory of spherically symmetric gravitational fields. The method
also applies to point masses embedded in an expanding universe and to general,
spherically symmetric black holes.
