Now I have exchanged the 340A for a DMOS transistor BS250. It was just lying around.
And now it works. The only lack due to its relative high R-DS-on is that the 5V on the board is only 4.6V.
But for my application the board is externally powered anyway. And then the 5V are really 5V.
I will try to find a transistor with a lower on-resistance than the BS250. But not something in SMD.
Do you know one?
By the way: A better controller than that one of the link above is completed. It has a connection via USB to receive motor impulses from the PC in synchronism with the captured camera frames.