They do, or at least some of them do.Is there a reason Irish banks don't allow you bring your mortgage with you
I know for example that KBC will let you move an existing fixed rate to a new property. I believe other banks will also, but it's the only one I've actually seen because I looked at it detail for someone.
It's just that with such a low penetration of fixed rates, and a history of reduced 'new business' rates, there's not usually much benefit to the customer (except for tracker rates of course)