We\'re using an API from a provider; in their next version, they would provide an OpenId server but mostly the users are using their desktop application. An other website we\'re
I think the answer is "it depends." For the close-and-reopened browser question, it depends on whether your sites use session cookies or persistent cookies to authenticate requests.
But as for the authenticate-with-desktop-client-and-then-open-a-browser, you'd have to ask the provider for more details. I can imagine ways to do it that would have either outcome.