While some individuals known for their expertise in rhetoric, philosophy, or music *might* travel to different cities to give lectures or teach privately, this wasn't their primary occupation in the same way a modern teacher has a salaried position at a school.
Instead:
* Sophistic teachers were a notable exception, often traveling between city-states offering their services in rhetoric and public speaking. They were more itinerant than other educators. Think of them as independent consultants or high-end tutors.
* Philosophers like Socrates, Plato, and Aristotle primarily taught in their own schools (the Academy, the Lyceum) within a single city (Athens). While they might have had students from other cities attending, they weren't routinely traveling to teach elsewhere.
* Educators in smaller communities were typically local and tied to their own city's needs.
So, while some learned individuals moved between cities and taught, it wasn't a common or structured profession like modern teaching. The structure of education and the social role of teachers were vastly different.