The most common and most likely answer to this crossword clue is the 7 letter word ALLEGRO
I've seen this crossword clue in The Times
A
L
L
E
G
R
O
More clues leading to the result ALLEGRO
- Everybody, say, runs over, moving quickly
- Completely self-obsessed Romeo breaks fast
- A couple of litres, say, run out rather fast
- Everyone, say, runs over briskly
- Claim with conclusion taken out and run over quickly
- Everyone, I must cuddle Republican quickly!
- Farfalle grocer hides quickly
- Fast or rising support for everyone, say
- Pacino goes to stage right, getting round quickly?
- A large block of building opened by king quickly
Buddy explains!

The straight part
We know that "briskly" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case briskly = allegro.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 ALLEGRO mean?
ALLEGRO noun- A tempo mark directing that a passage is to be played in a quick, lively tempo, faster than allegretto but slower than presto
- In more traditional use, an expressive marking indicating lively or happily, not necessarily quick or fast.
- A passage having this mark
- In a quick and lively manner
- Played in a quick, lively tempo