Beauty Dish, what fits what?

I know adaptor rings are made so you can use say something like an Elinchrom Octobox on a Profoto head but does this apply to beauty dishes as well.

The thing is, the Profoto Beauty Dish is 22" (I was told this weekend it's the holy grail of Beauty Dishes) and the Elinchrom is 27" (I was told it was a bit big but with a sock on it I loved the look) and someone elses was smaller.

Are you stuck with the one that comes from the manufacturer who made your light heads?

It depends really. The Elinchrom dishes don't have detachable mounting rings so they can't be adapted to other makes. Some of the best beauty dishes are made by specialist third parties (e.g. Mola) and can be adapted to nearly anything.

The Profoto dish is by no means universally held to be the best, and frankly it would be a waste of money to use it on a non-Profoto head as its focus isn't the same without the proper Profoto fitting.