I will transverse Russia by the trans-siberian railway in November. I will go all the way east to Vladivostok.
Then I want to go to Beijing.
Did anyone did that before?
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Rome2Rio says it can be done by bus via Harbin, China. From there you take a train to Beijing. The information presented by Rome2Rio is taken from the Lonely Planet getting-there webpage on Vladivostok which states:
Buses to Harbin, China, depart daily (except Sunday) around 6.30am (R2900, eight hours) from the bus station, 3km north of the centre. There are also frequent departures for Nakhodka (four hours) and other destinations in the Primorsky Territory. Some southbound destinations may be off limits to foreigners without a permit.
For more information about departures, China Train Guide has a page with the timetable of trains covering the Harbin - Beijing route.
From Harbin you can also take a bus all the way to Bejing. According to China Bus Guide the journey takes 12-13 hours, and the bus leaves once per day.