1. Each student is carrying a lunch box.
2. Each student is carrying lunch boxes to the nursing home.
(Does #1 mean as follows? Each student has a lunch box. #2 is different, right? In #2, each student is moving lunch boxes to the nursing home, right? I mean 'is carrying' is different in meaning in each sentence. Am I right?)
Yes, you're right.
in #1, the student (singular) has his/her own lunch box (singular).
In #2, each of the students is carrying or taking multiple lunch boxes, which will be distributed to several (or many?) people at the nursing home.