The most common and most likely answer to this crossword clue is the 4 plus 11 letter word ANTI IMPERIALIST
I've seen this crossword clue in Financial Times
A
N
T
I
I
M
P
E
R
I
A
L
I
S
T
Buddy explains!
The straight part
We know that "against old-style domination" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case against old-style domination = anti imperialist.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.