I am not sure - but it is a lot easier for a Broadway actor to go over than vice versa.
There is supposed to be recripocal agreements. For example, when American Equity did not want Sarah Brightman in 'The Phantom of the Opera' Andrew Lloyd Webber threatened to cancel the production (he had written it for her). In their subsequent agreement, he agreed to have an American actress in his next West End show. He actually had 2 - Ann Crumb and Kathleen Rowe McAllen in 'Aspects of Love.'
My mother always used to say, "The older you get, the better you get, unless you're a banana." - Rose Nyland