I've done quite a lot of work on the Funston family in England (it was my mother's maiden name) and can confirm that they can be traced back to a Simon Funston who is recorded as living in Newmarket, Suffolk in the 1470s.
The North American Funston diaspora, I suspect, has its origins in Ireland. Whether the English and Irish Funstons are connected I don't know. It's not an area I've yet looked into.
The Funston coat of arms almost certainly belong to one Thomas Funston who seems to have been a minor official connected to the Duchy of Lancester around the turn of the 16th/17th centuries.