From Longman Dictionary of Contemporary Englishrun around after somebody phrasal verb informalto do a lot of things for someone else as though you were their servant I’ve spent all day running around after the kids. → run→ See Verb table
Examples from the Corpus
run around after • In her shocked lethargy, she seemed content to let Valerie run around after her and do everything for her.• Now when he and Anne had people to stay they ran around after their guests making sure they were comfortable. 