Although built by the London and South Western Railway (L&SWR) this route to Wimbledon was also served by Metropolitan District Railway (MDR) trains from about the time the line first opened. Originally these were steam powered, and when electrification came (in 1905, I think) a system that would be compatible with trains operated by both railway companies (L&SWR and MDR) was chosen.
Until 1940 there was a shared service but war-time economies (ie: service reductions) saw what by then were Southern Railway 'all stations' services being withdrawn.
Until railway privatisation in 1994 this route and the two intermediate stations was vested in British Rail - even though the main service was as part of the District Line. In the past it was known for BR trains to call at intermediate stations on special occasions, such as when no District Line trains were coming this way, but nowadays the railway industry seems to prefer bus substitution - if anything at all. Of course the passengers would prefer trains, even if only a shuttle between East Putney and Wimbledon.
As previously stated, in addition to ecs (empty coaching stock / not in passenger service) trains travelling to and from Wimbledon depot some (nowadays South Western Railway) passenger trains still travel this way. I agree with those who see them as Parliamentary trains which are really about ensuring the legal fiction of all sections of a railway route still being in passenger service. Its not just for the connections at East Putney but also at Wimbledon.
What may also be noteworthy is that these trains are on longer distance services and they tend to travel at quieter times. The Real Time trains link above shows when they are running. An impediment to running a more frequent service is that only one track remains connected to the route to Waterloo at East Putney (the former northbound track that crossed over the Waterloo route on a viaduct has been removed).
For reasons I do not understand, when on this section of railway all mainline trains are muzzled at a somewhat sedate 30mph.
Simon