Usage is largely botanical. Can be used of leaves -- for example, those of some oaks turn brown but stay on the tree through the winter to finally fall in the spring -- or of blossoms -- such as of some squashes. The lower fronds of palm trees also comes to mind, which depending on the species can stay on for far longer than a pruner might desire. As for the word itself, it was borrowed in the 1720s from Latin marcēscēns, the past participle of marcēscēns, to shrivel, grow weak, the inchoative (an "about to begin" form) of marcēre, to wither, ultimately from the PIE root *merk-, to decay/die (which also gives us mortal).
The brown, marcescent leaves of stayed on her potted rubber plant long after she moved out, leaving it behind.