1
general::
Bring to a conclusion, carry out. For example, The second volume simply followed out the theories presented in the first, or He instructed them to follow out their orders to the letter. This idiom is dying out. [Mid-1700s]
American Heritage Idioms
2
general::
Phrase(s): follow someone or something out
to go out right after someone or something. • I followed her out and asked her if I could take her home. • The dog followed Billy out and went to school with him.
McGrawhill's American Idioms And Phrasal Verbs