The most common and most likely answer to this crossword clue is the 9 letter word SEESAWING
I've seen this crossword clue in The Times
S
E
E
S
A
W
I
N
G
Buddy explains!

The straight part
We know that "Swinging/inspector might ask to do this at a wood mill" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case swinging/inspector might ask to do this at a wood mill = seesawing.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does SEESAWING mean?
SEESAWING verb- To use a seesaw.
- (by extension) To fluctuate.
- To cause to move backward and forward in seesaw fashion.
- A rising and falling, like that of a seesaw.