Walk unsteadily, due to the snow-fall (7)

I believe the answer is:

stumble

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'walk unsteadily' is the definition.
'stumble' can be an answer for 'walk' (stumbling is a kind of walking). I'm not certain of the 'unsteadily' bit.

'the snow-fall' is the wordplay.
'the snow' becomes 's' (this could be a standard abbreviation which I don't know about).
'fall' becomes 'tumble' (synonyms).
's'+'tumble'='STUMBLE'

'due to' acts as a link.

Can you help me to learn more?

(Other definitions for stumble that I've seen before include "Fall while walking or running" , "Stagger" , "Trip and almost fall" , "Miss a step and risk falling" , "Falter" .)

I've seen this clue in the Evening Standard.

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