Yesterday my new bike stand has arrived. I celebrated its arrival by tackling my younger son's bike. Its rear gears wouldn't shift. I discovered that the shifter was producing all the necessary clicks, but the cable at the derailleur wasn't moving at all. Furthermore, I was unable to move the cable much - in particular, extract the outer from the fittings in the down tube (it's internal routed, without the outer inside the frame). From this I concluded that the lower part of the cable, the one that runs along the chainstay, has become rusted and merged with the outer. So I cut it in the middle, and then was able indeed to pull out the other half - it was indeed rusty through and through. I've just received a routing kit, but ended up doing without it, using the old cable as a guide for the new one. The new outer was cut to size, the new cable was greased and installed, all that remains is to adjust the derailleur.