Definitely swapping horses regularly. Being stuck on the same horse for months, or even years, leads a rider to developing bad habits and often leaves them finding it difficult to progress, or they progress slower and slower after a while on that horse.
By swapping horses, you are building up many more problem solving 'tricks' that you can use on other horses. You will encounter a greater range of problems and then when it comes time to buy your own horse you will find that you have more success with it than if you are stuck on the same horse your whole lesson life.
I've had a number of horses now, and still try to ride as many different horses as I possibly can. I will have the odd lesson on a school horse or on my coaches dressage horse, then also give people riding lessons and will get on their horses to demonstrate, I ride other people's horses to help them out with a problem their having, and will also ride other horses for the heck of it.
As a result, I have so many more tricks up my sleeve that I can resort to if I encounter a problem. I've ridden lazy horses, stubborn horses, hot horses, tense horses, forward horses, spooky horses..... and now there's not a hell of a lot that I see that suprises me or I don't have some kind of card to play to 'fix' the behaviour.
Fowl Play above mentioned the exact problem with people riding one horse. You get to know that horse which is great, but then you get thrown on another horse and hate it because you don't know what to do!! As riders, we need to be able to ride a number of horses SUCCESSFULLY!
Of course, as a total beginner being put on the same old schoolmaster for the first couple of months (depending on regularity of lessons and progression of the rider) is going to be beneficial to assist them in getting their balance and learning the very basics of stop go and turn. But once they're reasonably confident and competent, I am all for throwing them on different horses.
Absolutely definitely 100% I am with swapping horses around. It is the BEST thing for you as a rider, the added experience it gives you is just fantastic.