I don't think there's a specific word for that. Most of the people I asked said it was called self-contained classroom.
Some classrooms are self-contained, with one teacher and one group of students all day (or very close to it). US elementary classrooms are often like this, with a group of students staying together all day with one teacher aside from "specials" for something roughly like an hour per day 3-5 times a week. This is also used by some special education classrooms in higher grades.
Some schools use a cohort model, where students are with the same group of kids all day (with one or more teachers). I've heard that term used by US teachers in grades 6-9 (ages ~11-15). In some schools, students will move together as a group to the next classroom when it's time for the next subject, but I believe you can use the term if the teachers are moving as well.
Both of these words are kind of slippery because they can mean multiple things (e.g. a group of kids in the same grade can also be called a "cohort").
What is the term for schools where students stay in one class all day and the teachers change rooms?
I guess the classes in which students move from one class to another are called moving classes. And when teachers move from one calss to another, they're called moving teachers.
I've also heard rotating/ roaming teacher for the teacher who changes rooms.
After elementary school, students proceed to junior high school (also called middle school),
where they usually move from class to class each period, with a new teacher and a new
mixture of students in every class. Students can select from a wide range of academic classes
and elective classes.
Guide to the education system in the US