ashes, puckery juice, cremate
灰 centers on the powdery remains left after fire: ashes. The cremation sense is a specific application of burning to ashes, while "puckery juice" is an obscure, likely archaic meaning that does not fit the main conceptual center.
灰 combines 火 (fire) with a reduced form of 厂 (cliff), suggesting the remains of a fire under a cliff or shelter. The exact historical development is uncertain, but the association with fire and residue is clear.
Picture a cliff (厂) sheltering the remains of a fire (火): the gray, powdery ashes left behind. That's 灰.
For カイ, imagine a pile of ashes turning into a kite (カイト) and flying away: kite -> カイ, and the ashes scatter like a kite in the wind.
ash; ashes
lye
grey; gray; ashen
ashtray
volcanic ash
lime; quicklime; caustic lime
limestone
ash; embers; complete destruction
fall of (volcanic) ash; falling ash
lethal radioactive fallout; atomic dust
limestone
tuff
incinerated ash; burned ash; ash
heavy downpour of volcanic ash (in a short amount of time)
soda ash
soda lime
soda-lime-silica glass; soda-lime glass (most common type of glass)
being scattered in all directions; being broken up and dispersed
polyester bag for collecting volcanic ash
dust and ash