Is there any research supporting the idea that a single professor should teach the entire calculus sequence as opposed to splitting the duty amongst multiple professors? What are the pros and cons of having the same professor teach all the calculus classes?
Let me start with mentioning this now well-known experiment: Improved Learning in a Large-Enrollment Physics Class. I highlight part of which that is related to your question.
During week 12, we studied two sections whose instructors agreed to participate. For the 11 weeks preceding the study, both sections were taught in a similar manner by two instructors (A and B), both with above average student teaching evaluations and many years experience teaching this course and many others.
More or less, both lecturers used the same way to teach. And a variety of different data showed that both groups of students were similar in many different aspects (including learning) at the end of week 11. What happened at week 12?
The control section was taught by instructor A as before. But the experiment section was taught by two instructors with a very limited teaching experience using a more active teaching method as designed. What happened? The abstract tells it all:
We compared the amounts of learning achieved using two different instructional approaches under controlled conditions. We measured the learning of a specific set of topics and objectives when taught by 3 hours of traditional lecture given by an experienced highly rated instructor and 3 hours of instruction given by a trained but inexperienced instructor using instruction based on research in cognitive psychology and physics education. The comparison was made between two large sections (N = 267 and N = 271) of an introductory undergraduate physics course. We found increased student attendance, higher engagement, and more than twice the learning in the section taught using research-based instruction.
But, How all these is related to your question. Let us do a thought experiment. Suppose we want to design a research to compare what you are interested in. These are the variables that we should take the same: the materials covered and the order in which they are covered, the instructional methods used, the values (instructors' views of mathematics) and so on. Having kept all these the same, what remains to compare? Basically, nothing but some social aspects.
It is often recommended to attend a different university for graduate work than the institution where one did the Bachelor's degree. Why? To get a different perspective.
Extrapolating from that, I would expect variety to be helpful in the calculus sequence too.
Here's another point. Consider that different strokes work for different folks, but that not all math teachers know how to teach to a variety of learning styles. A student would therefore do well to hedge his bets and try someone different each semester (unless he finds someone he's remarkably well matched with, and decides to take whatever that instructor has to offer in a particular semester).
Finally, note that math is a bit like a foreign language. Those learning a language will do best in the long run if they are exposed to a variety of speaking styles. This would make sense in the math context as well.