I am not a school nurse, but have been around. There are some bb's for school nurses and you might want to go there and post your question.
do not allow students to routinely carry any meds, including inhalers, but wisely most have an exception policy. The schools I have knowledge of concentrate their school nurse power in middle and high school and so grade school kids are most likely to be seen in an asthmatic crisis by a school secretary or classroom teacher. The school nurse writes some sort of care plan for students with special health care needs, but I doubt that every kid with an inhaler even has one of these on him--if they do, it might be cookie cutter. I have known one nurse who did WPF's and she was using one cheap meter on many kids with different mouth piece (infection control problem). Legally, school nurses can retain responsibility for the care of kids even when they are not in the building; they are delegating med administration to the school secretary, classroom teacher etc. School secretaries often come from a special place in heaven and can be surprisingly good in this role, but if I were a school nurse my favorite nightmare would involve a child with status presenting to a school secretary when I was at another school across town.
Good schools have a nurse:regular ed student ratio of 1:700 (or the like); I've never seen that. So the main challenge of nurses is the educate their delegated care givers in the care of the chronic and acute asthmatic child, I think. Good luck. Obviously practicing school nurses will give you better info.