I did some research on this a number of years ago.
In the end I found that the FDS first mentioned by Jesus is later the one who was following the four horsemen of the apocalypse as detailed by John in Revelation 6:8. John refers to it as Death in that verse.
Some have suggested the FDS is actually riding on the Death horse but the scripture is unclear on that matter.
So whether the FDS was actually riding the horse or running alongside the horse is up for conjecture. Perhaps the fact that the horse was called Death may indicate the horse was sickly or nearing death so that it would be impractical to be riding on something that is very sick or weak.
While the details are not complete, we still can see the harmony that permeates the Scriptures.
Rub a Dub