So it looks like the main problem for this is trains going to elsewhere.
Would it be difficult for Northern to keep trains locally?
And UK wide which routes have trains stuck on a certain line/region?
I doubt anyone would mind a South Yorks livered train on Northern routes into neighbouring authorites as it will run there.
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)