The most common and most likely answer to this crossword clue is the 3 plus 4 plus 4 plus 4 letter word PUT ONES FOOT DOWN
I've seen this crossword clue in The Daily Telegraph
P
U
T
O
N
E
S
F
O
O
T
D
O
W
N
More clues leading to the result PUT ONES FOOT DOWN
Buddy explains!
The straight part
We know that "Move fast/to suppress dissent" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case move fast/to suppress dissent = put ones foot down.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.