The most common and most likely answer to this crossword clue is the 9 letter word RADIATING
I've seen this crossword clue in Financial Times
R
A
D
I
A
T
I
N
G
Buddy explains!
The straight part
We know that "Branching out" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case branching out = radiating.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 RADIATING mean?
RADIATING verb- To extend, send or spread out from a center like radii.
- To emit rays or waves.
- To come out or proceed in rays or waves.
- To illuminate.
- To expose to ionizing radiation, such as by radiography.
- To manifest oneself in a glowing manner.
- (ecology, intransitive) to spread into new habitats, migrate.