Wiktionary
get after
vb. 1 To move into action in pursuit of something. 2 To move into action in attempt to catch or defeat another. 3 To attempt to convince another to move into action.
Usage examples of "get after".
If they could just get the pope on one of those boats and downstream out of the city, they could retrieve the horses and get the hell out of Dodge a lot faster than any pursuit could be organized and get after them.
Then get after him, or I will hold you personally responsible, Dijmas.
I have written to Lestrade asking him to supply us with the details which are now wanting, and which he will only get after he has secured his man.