I'll share my experience, as well.
Context: As a graduate student, I was the lead instructor for a 120-student lecture course with 4 recitation sections amongst 3 TAs. Lectures were MWF and recitations TuTh. I assigned weekly homework assignments collected and returned in recitation. This was our "intro to proofs" course, a 100-level course entitled "Concepts of Math".
Method: I wrote and distributed detailed grading rubrics for homework problems. I also assigned specific problems to each TA, rather than have them grade everything from their own section. (Logistically, this was accomplished via a locked filing cabinet in a designated room that everyone had access to. Homework was turned in on Th, TAs would visit over the next few days to grade, and papers were returned on Tu.) My goal was to standardize grades across the sections throughout the semester, rather than at the end via statistical normalizing. My advisor had always used the end-of-term statistical method, but I worried this might foster jealousy of other sections with easy graders / contempt for TAs with high standards. I also helped out by grading one problem per assignment.
I used the same procedure for exams. I wrote detailed rubrics, including provisions for multiple methods of solution, predictions of common mistakes and how to handle them, and tips for providing good feedback beyond just a score. I graded a couple of questions on each exam, and read through a bunch of them to get a sense for how the students were doing.
For homework grading questions/complaints, I told the students to talk to their TA first, who would then direct them to the TA (or me) who graded the question. If that could not be resolved, they could come to me.
For exams, I gave the students exactly one week after return in which they could write up a document to explain their issue, their understanding of the relevant problem, and how their demonstration of knowledge should deserve more points. I would then read each of these and, when appropriate, raise the grade; either way, I wrote a response back to help them understand the grading as it was.
Results: I honestly think this was a great success, and am very proud of how the logistics of this course were handled, looking back. I never had complaints from students about unfairness, and I spoke with some of them often enough on a personal level that I like to think I could have determined whether there was an undercurrent amongst them that was just unspoken. When I did receive grading complaints, they were often of the form "I don't understand why I lost points here, can you explain more?" and I would do so on a case-by-case basis. Sometimes, their complaints were valid (and, indeed, there was one grader who was sloppy and unfair on early assignments, so I had to have a long meeting with them and show them how to better; thereafter, they improved greatly). Sometimes, they were just grade-grubbing, but in those cases, I was glad I had those formal rubrics written out, so I could point to the problem and say, "This part was worth X points, and you didn't address issue Y. There's nothing I can do, in fairness to other students."
Thoughts: As much of a success as this was, it required a huge time commitment from me as the primary instructor. I spent a lot of time designing the rubrics, taking into account my experiences grading similar problems in the past as a TA myself. I spent a lot of time answering TA questions, in person and via email, about grading issues. I spent a lot of time talking to students about homework problems (and how they were graded). I spent a lot of time just outright grading. But that was my choice. Ultimately, I enjoyed it all and don't regret it, simply for the experience that I gained from it all. I was glad to have TAs who cared as much about grading fairness as I did and didn't see it as just a teaching job on the side.
However, this is grossly infeasible. This was the one course I was teaching at the time while finishing my dissertation and, even then, it was cutting into my life. Like I said, I enjoyed all of it, but it just doesn't scale upwards nicely at all.
On the positive side, I now have a cache of homework/exam problems as well as their solutions and grading rubrics, and could happily share these with future instructors. So, perhaps this doesn't have to be a perpetual problem.
Recommendations: Frankly, I would strive for this but make sacrifices where they are needed to save time/effort. In the future, if I have more than one course like this, I would first cut the actual grading of problems myself. Or, at the least, I would make sure one problem is completely trivial to grade, almost like a multiple-choice problem. By reading through those submissions, I could at least get some sense of how the students are doing. I would then cut time in the development of the rubrics. As it was, I spent much time considering varied approaches to problems and trying to nip grading considerations in the bud a priori. This isn't necessary, if you're willing to put trust in your TAs and potentially spend the time answering their email questions as they come up.
Ultimately, I feel strongly that the statistical method is flawed, and personally prefer a method that attempts to standardize grading across sections. Specifically, I actively promote the each person grades a problem, not a section method. I believe this is the only way to be truly fair, and I don't think it requires that much more effort on the part of the instructor/TAs to be implemented.