You get a small paper at the entry, fill it out as of the following example. I got help of some locals, as it’s not in English.

Leave your bike on the right side and go into the building on the left with the pedestrians for the passport control.
After you’ve got your stamps (passport and the paper mentioned above), you can go back and get your bike and drive to the customs booth besides the building.
He will check the stamps on the paper and the passport. Now you can go the exit. There they will take the paper and let you out.

On the Kyrgizstan side you also have to leave the bike and go into the building on the left for the passport control. He spoke English this time, so no issues there.

They refused to give me a ‘temporary import’ paper. I still have the one from Russia, so there won’t be a problem to leave.