I'm not sure what you are asking. The non-headline form would be. "A suspect is on the run after a man was found gunned down."
Student or Learner
"Suspect on the run after man gunned down"
In the above heading, which word is the phrasal verb "gunned down" followed by with? Is it connected with "man" or "suspect"?
The man was gunned down by the suspect.
How can it be the suspect shot him down/killed him?
I am not a teacher.
Now that the OP's question has been answered I would like to add a comment regarding the over-use of the word 'suspect'. Under certain conditions there may be legal reasons for referring to people as suspects, but when legal constraints don't apply, calling a perpetrator a suspect smacks of political correctness.
To me, a suspect is someone whose guilt has not been established.
In this case a man has been gunned down by someone. A person suspected of the crime is on the run. Nothing wrong with that.
The man was gunned down by the person that gunned him down.