BTW, the biggest meikil on mamzerim in the US was R’ Moshe Feinstein. R’ Henkin (who was THE gadol hador in the US) rules that a Conservative (maybe even Reform) marriage requires a get. R’ Moshe holds that they don’t need a get.
According to R’ Henkin, those children from second marriages would be mamzerim, since the first marriage is still valid.