The most common and most likely answer to this crossword clue is the 7 letter word SWIFTLY
I've seen this crossword clue in The Sunday Telegraph
S
W
I
F
T
L
Y
Buddy explains!

The straight part
We know that "with expedition" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case with expedition = swiftly.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 SWIFTLY mean?
SWIFTLY adverb- In a swift manner; quickly; with quick motion or velocity; fleetly.