What happens if a professor's interests shift dramatically?
Assume that the change is gradual, not abrupt. Alice will want to learn something about the new field so that she is able to do effective research in it. No new department is likely to accept her until this happens. She is also unlikely to completely abandon the old field due to the many constraints, both financial and ethical. Yes, she needs to see to the needs of her current students somehow, and may need to keep publishing in the old field if she wants good annual evaluations. 

Once she has a few nice publications in the new area she can start to look for a new position. She probably doesn't need to get any additional degrees unless that is needed to obtain the expertise in the new field, but it would probably help her job search to have some formal qualification in the new field. 

One way to get some expertise is to try to collaborate with experts in the new field, first as a junior partner. It is best, of course, if she has something to offer her collaborators, perhaps some research perspective from the old field. 

For smaller jumps, of course, it is much easier. Many people switch back (and forth) between math and computer science, for example. This is true even though the basic research methodologies in those two are quite different. I switched from math to CS over a few years at a time when the institution I was at needed expertise in a developing field (long ago). But a person who switches here likely is able to use many of the ideas in the "old" field to enhance their work in the new.