If the gargoyle died as flesh and blood, he or she is cremated; if he or she died during stone sleep, his or her remains are pulverized (a disturbing concept, but gargoyles very seldom die in their sleep - except when shattered). The mourners then take the ashes or dust to the highest point in the local area, and hold a memorial service on the departed; by gargoyle custom, everyone is permitted a voice, even the enemies of the deceased. At the end, the gargoyles spread the mortal remains upon the wind, with the words "Ashes to ashes OR dust to dust. All is one with the wind." (This ritual speech was later adapted and used in Western human funerals). The gargoyles then glide out amid the wind-borne ashes or dust, so that part of the deceased will remain with them forever.
After the Sruighlea Cell of Demona's Clan was massacred by Constantine and his men, "True" suggests that the clan perform a Wind Ceremony for the lives lost. Demona suggested instead that they kill those responsible. It is unknown whether any gargoyle returned to the cave to perform a proper Wind Ceremony for the gargoyles killed there. ("Tyrants")
It has been suggested that part of the reason why Xanatos and Demona were able to revive the Coldtrio through their creation of Coldstone was that the Wind Ceremony was never performed for them after the sack of Castle Wyvern - a concept somewhat reminiscent of how, in Greek mythology, the souls of the dead who were denied funeral rites were unable to gain complete access to the afterlife in the Underworld - cf. Homer's Iliad and Sophocles' Antigone.
Any gargoyle eggs that do not hatch due to "stillborn" or "bad egg" issues are also given a Wind Ceremony.
Despite some similarities, the rest of the clan standing around what they believe to be Hudson's remains in "The Price" is just the early stages of mourning, not an actual Wind Ceremony which would have come later.