There are probably examples of all of these, and more.
The variety of bus drivers needing to travel between depot and change-over point (or whatever the local terms are) has probably increased over recent years with a trend towards selling off town centre depots / garages and having fewer operational sites, many on 'out of town' industrial sites.
London bus operators' staff get oyster cards which allow free travel on all TFL services, so there are plenty of examples where bus staff use TFL bus routes (irrespective of operator) and Underground (and possibly Overground / TFL Rail) services. Some can be quite unexpected - I believe that London General drivers from Stockwell Garage have used the Victoria Line to get to points including Warren Street and even Finsbury Park for a few duties when they ran route 19.
While on the face of it this is 'inefficient', it's a balance between 'unproductive' travelling time and making the duty schedule work - you sometimes get duties that are mostly too short (most operators have a guaranteed minimum payment either per day or per week, so short duties cost money) or would be just too long for drivers' hours. The occasional driver change at the 'wrong end' of a route can remedy this.
Some change overs can be done by the driver who's due a break / end of duty taking bus back to depot and another driver bringing a bus out, but this adds to bus mileage, and unless the times all come out off peak (and most bus routes have fewer 'spreadover buses' (peak hours only) than they used to), it can add to the 'peak vehicle requirement' (number of buses you need to have.)
A number of operators have staff cars (terms for this vary - staff cars, shuttle cars, buggy etc) that drivers use to get from A to B. This can be cheaper than needing extra buses to do it, or relying on having spare buses available. Arriva at Dartford (and potentially other places) used to have a fleet of ex London taxis for this, but think they have fallen by the wayside -
this car on Street View at Dartford Home Gardens is parked on a bus stop, and there's glimpses of what looks like a high-vis and an Arriva blue shirt visible inside the car, so it's probably one.
Letting the bus drivers drive these cars works if the driver who's getting off a bus will always be going back to the depot where the relieving driver has travelled from, but if the bus drivers take their meal break at the remote location, that can mean you end up with needing to park those cars somewhere, which may not be possible at bus stations.
Go-Ahead London have staff cars with dedicated drivers (no specific inside knowledge, I've seen job adverts for the staff car driver post) at some locations. Northumberland Park has a few, there's one visible at Tottenham Hale on Street View
here (the red MPV) - presume the drivers get taken to Tottenham Hale Station and then get the Victoria Line to Walthamstow (for routes that do change overs at Walthamstow.) I've also noticed similar at North Greenwich, presume shuttling drivers to / from Morden Wharf Garage. I'm sure they have done the sums and decided this is the best way of doing it.
Away from London, I'm aware some Reading Buses drivers on the Green Line (or whatever it's called now) 702 between Legoland and London do change-overs at Slough and travel at one end of the duty between Reading and Slough by train.
Reading Buses also have a minibus that loops round Reading Town Centre in the early evenings showing 'shuttle bus' - this isn't a public service, and the only people I've ever seen getting on or off seem to be Reading Buses drivers, so presume this is for getting drivers to / from their depot, which is at the edge of the town centre. I've not seen it running daytimes, it's possibly only run in the evening.
I also have a vague memory of seeing an Arriva minibus running to and from Aylesbury Bus Station showing route number 999, which looked only to be carrying other Arriva staff as passengers - this was a few years ago now.
