Transferring a visa to a new employer in another city

Hi! I'm currently working in Hangzhou on my first ever visa in China. My job is lame, and I'm trying to change careers from teaching to editing copy and producing educational materials. I am applying for jobs, almost all of them in Beijing or Shanghai. I have a few who seem interested but I'm worried about the next step scaring them away.

What do I need to do in order to transfer my visa? I want to be prepared and make the transition as smooth as possible.