The most well known ones from my point of view is Northern Carlisle and Newcastle based crews covering the scotrail Glasgow-Kilmarnock-Dumfries-Newcastle workings (a scotrail 156 overnights at Heaton and works ECS from Newcastle-Heaton coupled up to northern units) along with the TPE crews and units covering northern services out of Barton. I think that although northern have more crews now working the Yorkshire coast the first and last services are still covered by Scarborough TPE crews using northern 158s as a unit stays in the station overnight but I’ll have to check with one of my scarbados colleagues to find out. Freightliner crews working TPE Anglo Scottish work has now ended now TPE have their own Glasgow based crews to work both ECML and WCML services out of Edinburgh. they may still be used for route conducting duties along the (albeit rare) S+C, GSW and Doncaster-York ECML if required, this arrangement may be ending tho as York drivers are now signing York-Doncaster IEP for ECS moves and should NPR work require diversions it is likely York, Newcastle and Sheffield conductors/train managers will also sign this missing link. With the departure of Tim Brawn from TPE 3 years ago there sadly hasn’t been any BLS charters organised since which also saw freightliner crews conducting TPE staff well and truly off the beaten track
The GC 180 plus crew working a morning commuter service into Leeds won’t be running at the moment as GC have suspended their operations
I wondered about the ScotRail services to Newcastle a few years ago, and was fairly sure they changed operator at Carlisle. I've just checked the GBTT for 2000, and
all Tyne Valley services - including through trains to and from Dumfries, Kilmarnock, Glasgow, etc., are shown as operated by Arriva Trains Northern. More recently, but still a few years ago, when travelling with a special Northern-only day ticket, I planned to catch the evening Glasgow - Newcastle from Carlisle, so checked the operator: RTT showed it changed to being a Northern service at Carlisle (and the Northern crew accepted my ticket without question). This would mean that this was actually ScotRail providing a unit to Northern, rather than Northern a crew to ScotRail! It's strange that this situation was set up when the railways were split up for privatisation, but I guess that because the service was already there it had to continue and it wasn't felt right for one ex-Regional Railways TOC to be running services on a route for which a different ex-RR TOC had responsibility.
Doing some digging, it appears almost all the through services have been withdrawn. In the current non-emergency timetable there are two Dumfries - Newcastle workings early in the morning and the aforementioned Glasgow - Newcastle in the evening, but nothing the other way. RTT shows these all as ScotRail services throughout; however, Charlwoodhouse Live Rail shows them having a change of service code from 23579003 (ScotRail - G&SW routes) to 21793000 (Northern - Tyne Valley). Now I have to confess I don't fully understand the technicalities of all the codes, so this could simply mean it's crewed by Northern. But I then went back a year to see if I could work out when the more frequent through trains stopped running...
Until the May 2019 timetable change there was also a Newcastle - Glasgow Central in the morning, presumably using the unit which had been at Heaton overnight. Charlwoodhouse Live Rail still (just) has schedules from the December 2018 to May 2019 timetable, and shows this train as being a
Northern working, but with a change of service code (21793000 to 23579003) at Carlisle, therefore reinforcing my belief they have always been Northern trains east of Carlisle and ScotRail trains north of the border city.
Incidentally, it appears that in the current "Coronavirus" timetable only one of the through services runs. This is the first of the two Dumfries workings. Tomorrow, it's running with an STP schedule as a Northern train throughout, with the WTT schedule cancelled. However, RTT shows the cancelled WTT schedule as "NT" on its Carlisle listings, but with the train shown as operated by ScotRail when you look at that page!