The troop train running Edinburgh to Liverpool that was involved in the Quintinshill accident in 1915 was formed of Great Central stock. There were payments from one company to the other for the use of stock, but commonly the balance between any two would be adjusted over time to even out without payment either way. The Railway Clearing House managed this - and for freight wagons even more so.
The Somerset & Dorset Joint had its own locos (and numbering sequence) and carriages, in its own blue livery, until 1930, when losses led to the end of independence, there was a division of the various responsibilities between the two owners, the locos thereafter being LMS types and the carriages being supplied by the Southern, so all their local trains were mixed ownership. This lasted right through, with LMR locos and SR green stock, until the line closed in 1966.