  The instability against emission of fermionic particles by the trapping
horizon of an evolving black hole is analyzed using the Hamilton-Jacobi
tunneling method. This method automatically selects one special expression for
the surface gravity of a changing horizon. The results also apply to point
masses embedded in an expanding universe. As a bonus of the tunneling method,
we gain the insight that the surface gravity still defines a temperature
parameter as long as the evolution is sufficiently slow that the black hole
pass through a sequence of quasi-equilibrium states, and that black holes
should be semi-classically unstable even in a hypothetical world without
bosonic fields.
