I found the answer by browsing around. It seems that this is a long-time problem, caused by Sony's deficient software and their poor description of how to use it.
The "auto" function of the USB connect on the camera doesn't work properly; one must set this to MTP. Then connection works fine, in both directions, in the PlayMemories software.
Why it goes via a USB setting, when USB is not involved, defies understanding but that is the way it is!
Kit: all Sony now:
RX-100/3, A6000, A7R. Lenses:FE16-35, FE24-70, SonyF10-18,SonyF18-200, FE35/f2.8, F16/2.8, Minolta APO 100-300, Nikon 135 macro, Nikon 50/1.4, various extras.