I'm at a small mom and pop, but we don't normally "hand folks back and forth". There are times when I'm out of town that I'll ask another instructor if they would mind flying with my students while I'm gone. Some students are OK with it, others would rather just wait until I get back. In our FBO office there is a binder for each instructor. In the binder there is a divider with a syllabus for each of their students. Even though they could fly with different instructors, there is always a "primary" instructor and they are the ones who will solo the student or sign off for a checkride, etc. (unless the student would rather change primary instructors). There are different tasks with lines for student initials, instructor initials, and the date it was completed. Some tasks are not complete until the lesson is satisfactory and the student is proficient (steep turns), others are just a completion (night work). There are probably better methods out there, but this works for us.