When I was TAing a course, some students came to meet me to talk about marking issues. Earlier, the instructor had said that if students cannot reach an agreement with TAs they should talk to the instructor. Thus when I saw some students disputing marks, I referred them to the instructor. However, the instructor later sent me an email saying he couldn't decide the marks as the TAs made the marking scheme. In my view, he somehow refused to fulfill his duty and passed the buck to me.
Generally, unless the mark is extremely, obviously wrong, the professor should support the decision made by the TA. However, this means that when there is a grading dispute you cannot resolve with the student, you need to refer the student to the professor, but also talk to the professor yourself, explaining your marking scheme and reasoning behind the grade. Then the professor is equipped to explain to the student why they got the mark they deserve. You cannot expect the professor to moderate a dispute unless they actually have the grading scheme and your side of the story.
Also, you shouldn't immediately pass the student on. Let them present their case to you first, and explain to them why you assigned the grade you did. Nine times out of ten, the student is happy with that. You should only refer the one out of ten who refuse to see reason to your professor.