The most common and most likely answer to this crossword clue is the 7 letter word PASSANT
I've seen this crossword clue in The Times
P
A
S
S
A
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More clues leading to the result PASSANT
Buddy explains!

The straight part
We know that "Walking in a particular direction" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case walking in a particular direction = passant.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 PASSANT mean?
PASSANT adjective- (of a four-legged animal) Walking, usually to the right, and looking straight ahead with the right forepaw raised from the ground.
- Currently in use; in vogue.