They relocate every time you head through them. The rings vary in color. Different colors mean a different amount of beans will be given. As a side note, you cant dismount the broomstick unless you land at the Hogwarts front entrance, or in front of the Quidditch Stadium. Just press the button again and Harry will dismount.

Another way to find beans is to blast every tangible object with Flipendo. Most objects will cough up at least three beans. When you return to them a later time, they sometimes will give up more! Objects like masks, vases, jugs, boxes, etc. should all have some beans to share.