Drops off, divines, runs inside (7)

I believe the answer is:

drowses

I'm a little stuck... Click here to teach me more about this clue!

'drops off' is the definition.
(I've seen this in another clue)

'divines runs inside' is the wordplay.
I cannot quite understand how this works, but
'runs' could be 'r' (cricket abbreviation) and 'r' is found within the answer.
The remaining letters 'dowses' is a valid word which might be clued in a way I don't see.
This may be the basis of the clue (or it may be nonsense).

Can you help me to learn more?

I've seen this clue in the Sydney Morning Herald.

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