Can failure of an old electrolytic capacitor be precipitated by motion?
It's very unlikely that an electrolytic capacitor failed as result of being moved. They generally either degrade gradually as the electrolyte dries out (greatly accelerated by high temperature) or they fail catastrophically and vent or bulge, particularly when abused or badly made.  

Much more likely is some kind of mechanical damage, a solder joint or PCB cracking or a connection becoming dislodged. If it's an unpackaged board, the possibilities of mechanical damage are much greater- from missing configuration jumpers to surface-mount components completely sheared off. 

And if the board is not enclosed (sometimes even if it is) there is the possibility of ESD damage, especially at this time of year in the Northern Hemisphere when humidity levels tend to be low.