Progressed in a graceful curve in the warm sea (7)

I believe the answer is:

marched

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'progressed in a graceful' is the definition.
I can't judge whether this definition defines the answer.

'curve in the warm sea' is the wordplay.
'curve' becomes 'arch' (synonyms).
'in the' means one lot of letters goes inside another.
'warm sea' becomes 'med' (short for Mediterranean. I am not sure about the 'warm' bit.).
'arch' going into 'med' is 'MARCHED'.

Can you help me to learn more?

(Other definitions for marched that I've seen before include "Walked in step" , "Walked with measured strides" , "Moved with measured strides" , "took part in moving protest" .)

I've seen this clue in the Evening Standard.

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