I struggle a lot with the idea of a late work policy. In a purely philosophical sense, I don't believe in charging penalties for late work--because if a student does it, and the purpose is to learn, then there shouldn't be a penalty for when the info is learned.
However, we teach in the real world, and grading 2500 assignments at the end of a grading period just isn't feasible. And we all know that assignments turned in at the end of a marking period tend to be facile and completely, well, lame. This is because the work is done out of context: if you're writing, for example, about a text you read 3 months ago, then you're not going to remember all that much about it. Your analysis is going to be cursory at best and, more likely, just flaccid and dull and a waste of space. Also, as teachers, we know that we tend to grade assignments cursorily in the end-of-semester crunch as well, so not much actual learning happens.
What I'm toying with now is the idea of chunking the semester into 2 or 3-week blocks, letting students turn in whatever assignments they want, whenever they want within that time frame, but not afterwards. Obviously, major projects and presentations have to be done on time, but the rest will allow for students to move a little bit at their own pace, but not get super far behind. Hopefully, it will serve to make my ends-of-semesters easier as well.
This is a thought-exercise in progress, and would love to hear what your additions + suggestions are.