Stamp out what the Cheshire Cat did (3,4)

I believe the answer is:

die away

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'the cheshire cat did' is the definition.
I can't judge whether this defines the answer.

'stamp out' is the wordplay.
'stamp' becomes 'die' (I've seen this in other clues).
'out' becomes 'away' (synonyms).
'die'+'away'='DIE-AWAY'

'what' is the link.

Can you help me to learn more?

(Another definition for die away that I've seen is " slowly fade out".)

I've seen this clue in The Times.

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