I can only speak for the division of a large group I have worked for, and an independent. I suspect it varies a lot though, depending on the distances between depots, staffing levels, sizes of depots, and other variables etc.
The large group operator would typically just be drivers moving stuff as required on overtime or a "spare" driver running it over (or more often a "spare" driver cover the shift of a chosen driver). If it was a case of several vehicles moving to another depot but not an equal number coming back, then either someone would follow in a ferry car and bring the group back, or if just one/two vehicles sometimes get a train back (or vice versa if picking some up but not taking anything in exchange). That very much varied depending where the vehicles were going obviously. If it was a swap of a similar number of vehicles then drivers would just take them over and bring the "new" ones back. Occasionally they'd be tied in with rail replacement work, but this only happened a few times and was generally quite coincidental. For example Depot A wanted to swap some vehicles with Depot B, so Depot A put them out on rail replacement work one weekend that happened to be not too far from Depot B, so at some point during the day the vehicles would be swapped (in one case the end of the rail replacement work terminated at Depot B's nearest rail station so on a layover the buses simply drove to the depot and the driver took the swaps back out). The drivers had to inform the controlling rail replacement organisation of the vehicle swaps though as they like to know which vehicles are out, and obviously requiring such luck of everything falling into place was extremely rare. New vehicle deliveries from Alexander Dennis were generally delivered by drivers from the manufacturer, rather than our drivers going to the factory to collect them. Repaints were done in-house at another depot, so once a batch was started, that was generally a driver take one for repaint and bring the repainted one back, and repeat until the batch done. Sometimes fitters would transfer the buses themselves, sometimes with someone following in one of the engineering vans to bring their colleague back. A lot of our swaps seemed to get done on Sundays when work was lighter.
The independent I have worked for was only very small, so was generally one of the bosses doing it. For example catch a train a dealer to collect a purchase and drive it back themselves.