A new technique is used to study a family of time-dependent null horizons, called "Evolving Null Horizons" (ENHs), of generalized Robertson-Walker (GRW) space-time [Formula: see text] such that the metric [Formula: see text] satisfies a kinematic condition. This work is different from our early papers on the same issue where we used (1 + n)-splitting space-time but only some special subcases of GRW space-time have this formalism. Also, in contrast to previous work, we have proved that each member of ENHs is totally umbilical in [Formula: see text]. Finally, we show that there exists an ENH which is always a null horizon evolving into a black hole event horizon and suggest some open problems.