Assignment due dates
I would recommend not batching up assignments to a common deadline, because that makes it even more challenging for students to manage their time properly. If someone waited to the last minute on an assignment in the prior regime, now they'll be waiting until the last minute on 3 assignments and be in even worse shape -- compounded by the mental effort in prioritizing which one they should work on first. Under the original regime, they have more opportunities to engage in this cycle and get feedback on how their study skills work for them. 

My response to these types of problems is to have a blanket policy of dropping a certain number of grades from consideration; for example, in my classes, there are 12 weekly online quizzes, and the lowest 3 are dropped for each student at the end of the course. There are absolutely no makeups or past-due work allowed for any reason, which massively clears out my own time (absolutely no chasing students or managing late work after my grading process is completed). Plus, the "drop N grades" is automatically supported by the Blackboard management system. 

In my classes, I don't give any special allowances to students who first show up later in the semester (but this may vary for your institution). I had a student join my class in the 2nd meeting last week, and he had already missed the first quiz submission; he asked "Can I make that up?", and the answer was simply, "No", and being pointed to the syllabus for the drop allowance, which in the years I've been using it has always been accepted quite gracefully. In my opinion, it's the most efficient use of your time as an academic.