Had a moment to test it more. Surprisingly problem is not related to cartridges (your theory seemed right) because mechanism locks even when model is empty :-/
There are two main moving parts in my Winchester mechanism: silver "carrier" that travels up and down to transport cartridges from tube magazine under barrel, and black bolt that pushes cartridge from carrier into the chamber and then extracts empty cartridge over carrier.
Lockage occurs when lever is moved down: carrier move down, while bolt moves back. It is near end of this cycle that problem occurs. As far as I can tell carrier does not reach bottom position, but stops about 1-2mm before reaching it. This means that front face of the bolt, which is wider, cannot clear top of carrier and gets locked on it.
Moving lever up/down does not necessarily help, as the carrier either does not move at all, or finishes in the same, slightly misaligned position. It seems that only when lever is moved very energetically, carrier goes all the way down giving enough clearance to bolt.
Next step: disassemble, check what pushes carrier down, see if it can be made to push it further. Maybe this part is worn, or carrier hits some obstruction on the way down...