GA flight instructor with student reported an NMAC during approach to DYB non-towered airport. The conflicting aircraft pilot was not appropriately communicating on UNICOM; which contributed to the event.

Narrative

Was already established in the pattern in the crosswind for right traffic. Immediately received a traffic ahead notification from the G-430 [Garmin 430]; immediately turned into the downwind with steep turn. The traffic was straight ahead about 400 feet above our aircraft. Did not hear anything on UNICOM; did not know if aircraft intended on entering the pattern or if they were flying too close to the airport. Established my downwind call for right traffic. Immediately heard on UNICOM; Aircraft Y entering the 45 right downwind 6. Was losing altitude and coming right for our aircraft. My student kept saying they are getting closer. I saw on the ADS-B they were our same altitude at 1;000 feet for traffic pattern. Aircraft Y continued to fly right towards us. I immediately dove down to get away; unfortunately Aircraft Y was also descending. I leveled off at 500 and made an immediate left turn about 90 degrees; started to climb once I was clear of the other aircraft. Heard on the comms left base for 6 by Aircraft Y; but it is right traffic pattern. They did a go-around and departed the pattern. I look back and wish I had said something on the comms to make sure we stayed clear of each other; but in the moment I was concerned about getting down and the safety of my student.