Wiktionary
run after
vb. 1 To chase. 2 To make a determined effort to win someone's affections. 3 To endeavour to find or obtain.
Usage examples of "run after".
She thought more than once to call him back or to run after him, but she knew he would not want that.
Regaining my composure I saw that two of the younger boys had run after my suitcase and were lugging it up the path toward the house.
She turned back to Jarrett, stopping him when he was ready to run after her.