With the 'mother breed' we have 3 classifications.
First there is the Braunvieh, the breed in switzerland
Then there is what we Call 'Original Braunvieh, which is an American breed taken from the Braunvieh, bred for beef (but still decent for milking)
Last there is the Brown Swiss, which is another variation of the Braunvieh, this bred for milking.
As for crossing, it has happened in all directions.
Swiss breeders have crossed some American lines into the Swiss Braunvieh stock, American breeders have recently imported new blood from Switzerland and crossed it into the local Brown Swiss Stock, and Original Braunvieh have been crossed both ways. Really all this is, is that farmers are getting different bloodlines. American Brown Swiss are descended from a few cattle imported up until the 1880s. Swiss cows carry on those and a number of other bloodlines. American bloodlines have maintained a strict pedigree from a certain line, because that is all that was available. Swiss breeders got hold of some of this line, and crossed it back into the parent stock.
Sounds complicated, that's how breeding works. We're always crossing lines back into each other. With humans, we'd call it inbreeding