Crossword Genius

Go to and fro out West, say (4)

Ross

I believe the answer is:

sway

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'go to and fro' is the definition.
The answer and definition can be both acts as well as being singular nouns.
Perhaps you can see an association between them that I don't see?

'out west say' is the wordplay.
'out' indicates an anagram.
'west' becomes 'w' (synonyms).
'w'+'say'='wsay'
'wsay' with letters rearranged gives 'SWAY'.

Can you help me to learn more?

(Other definitions for sway that I've seen before include "Influence; dominance" , "Bend; influence" , "Lean from side to side - influence" , "Move from side to side; influence" , "Oscillate" .)

I've seen this clue in the Evening Standard.
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