According to club historian, Nigel Williams in his book 'Bradford Northern - the History 1863 to 1989', the RA&B came from the Bradford Cricket, Athletic and Football club which had built the park Avenue ground in the 1870s and with which the earlier [dating back to 1863] rugby/football club [the terms were interchangeable, back then apparently] and were playing at Apperley Bridge, merged with in around 1879 to form the Bradford Football club. Which then went on to split and become Northern and Avenue, etc......
No real idea about Dewsbury, but given the Bradford history I doubt there is any connection.