Hurrying from place to place astride galloping hunter (2,3,3)

I believe the answer is:

on the run

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'astride galloping hunter' is the definition.
(I've seen this before)

I cannot understand how the remainder of the clue works.

Can you help me to learn more?

(Other definitions for on the run that I've seen before include "one after the other" , "In flight as from the police" , "retreating?" , "Where bobsleigh competitor is found" , "Like fugitives" .)

I've seen this clue in The Telegraph.

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