Are you mated for life when you enter a school or can you change from one to another? Can you listen to more than one instructor?
From a mechanical point of view, switching schools is usually not a big problem. Lots of people switch from one to the other with relatively little problem, and at the upper levels of the Kendo Federation (as folks get older and sensei pass away) there can even be Muso Jukiden Eishin Ryu instructors who are students of Muso Shinden Ryu sensei and vice versa.
Most of the difficulties in changing from dojo to dojo are to do with koryu and other small organizations based on similar principles of one on one teaching. It’s a lot easier to change dojo or instructors, even arts, in a large organization than it is in many koryu. This is due to several factors, one may perhaps be a somewhat less “proprietary” viewpoint toward students. When there are several sensei who regularly interact with each other it’s not hard to pass students along if they move or even if they just need a bit of attention from someone else. Another factor is the sheer number of sensei, when you have multiple dojo in the same region it’s not too hard to have a student population that becomes somewhat transitory, practicing on Monday and Wednesday at one dojo, Tuesday and Thursday at another. Once established, this will create a situation where the sensei can’t very well forbid students from looking at other sensei even if they decide they want to.
However, even in a classical Koryu situation it’s sometimes necessary for students to take a good long look at their sensei, whether or not they are moving to a different place. A sensei may become unstable, egotistical or otherwise dangerous, he may decide to leave a wider organization, or he may simply become stale and uncaring about his instruction of the students.
On the other hand, a senior sensei may retire, die or leave the area, and the incumbent sensei may be junior to the remaining students, just not very skilled, or not capable of maintaining a group due to poor people skills.
And finally, a student may become “deaf”, incapable of learning more from the sensei.
Ultimately it is the student’s responsibility whether or not they stay or move. As far as I know there is no mechanism to “blackball” students beyond the usual gossip network of instructors that has always existed, so if you leave one, you can usually find another to take you in.
In an ideal world of course one would not ever need to change one’s sensei but that sort of wishful thinking isn’t very budo is it? We need to live in the real world of the simple complexities of dealing with other people.
Students must do what’s best for them, for those who come after them (their own students), and for the art if they are inclined to care about that, all the while guarding against decisions that are derived from their own ego.