In a similar vein to the suggestion by Chris J on late submissions, one idea to late assignments is:
Rather than having assignments close after the due date and thus having students not be able to access them until the teacher manually extends due dates, one idea is to simply always keep assignments open, but if a student completes it after the due date, a zero is transferred into the gradebook. This way, a teacher can tell a student that he/she can always complete late work-- but it will only be accepted based on criteria set by the teacher (e.g. doctor's note, etc.). The advantage of this is that there is less work for the teacher because he/she doesn't have to manually change due dates and students don't have to wait for the teacher to do this in order to complete late work.