As someone else said it isn't usually possible for a locally liveried train to stay on the same route.
You see this all the time on the buses, and IMO it discourages people from using them - seeing a bus advertising the Barnsley-Doncaster-Airport route on a local Doncaster-Scawsby route is only going to confuse folk who don't normally use a bus - which is correct? The branding or the matrix display? Stagecoach are the worst for this locally, First slightly better - but First South Yorkshire have a dedicated fleet of hybrid buses in Sheffield which they keep for certain routes there. The plebs in Rotherham and Doncaster don't deserve them!
Anyway . . . trains . . . !
As a non-regular train user if I saw a train with a local-to-elsewhere branding on it I would be hassling the train staff before boarding to ask if it was actually going where the announcements stated - especially when some Northern services have very few / no destination boards. (Was on a 322 from Leeds to Doncaster a few weeks ago which had no destination sign on, no stop announcements, no indication of where it was going other than platform displays)