Applying for a Visa while traveling

Hi there, My girlfriend and I (UK passport holers, with a working/residence type in Korea) will be leaving from South Korea and travleing around Asia, then want to go into Beijing and get the trans Siberian. I'm having trouble finding out how to apply for a visa in another country / 3rd country.
Most of the sites say you must have a valid one to live / work in that 3rd country to apply for a Chinese one. We have applied for many visas to PRC from S.K but they were dependant on having a S.K residence type.

Is there any way to apply for a Chinese one without having a working/residence type in the3rd country of application?
Many thanks
Phill