Yes, put the driver up for sale from team A and from team B go and buy him. But there are few issues with this for instance if the driver level is higher than the manager level of team B then you might not be able to purchase him, when you put him up for sale from team A there is a risk someone else might purchase him so you will have to be quick in buying with Team B, I can't think of anything else so yeah.
But there is no other direct or simpler way to swap driver or staff between teams.
