C172 Flight Instructor reported a NMAC event during landing pattern training when an aircraft entered pattern and joined into the number one position for landing. The Flight Instructor executed an evasive maneuver to avoid a collision.

Narrative

My student and I were conducting a local training flight in the vicinity of ZZZ. The airport conditions were VMC and many local training flights were occurring; creating a very congested traffic pattern. In compliance with FAA and company recommended procedures; Aircraft X was well established to enter the pattern on a correct 45-degree pattern entry to the left downwind for Runway XX at ZZZ and routinely made common traffic advisory frequency (CTAF) position reports. After aircraft one made a position report for a 3-mile 45-entry to Runway XX; the other aircraft (Aircraft Y) proceeded to make a call stating they were entering the 45 for the runway. Aircraft X did not have a visual on the aircraft and inquired Aircraft Y to state position and advise if they had the 45 traffic in sight. After inquiring multiple times; Aircraft Y stated they had Aircraft X in sight and wanted priority to land because they had been flying in excess of four hours. Aircraft X followed FAA-recommended procedures to enter the left downwind and made a position report that it had entered the left downwind for Runway XX. Less than a minute later; aircraft two made the same radio call stating they entered the pattern at the mid-field point; and Aircraft X was able to establish visual contact with Aircraft Y. Instead of entering on the 45-degree entry; Aircraft Y proceeded to enter the pattern at the midfield left downwind point and make a 90-degree turn to join the downwind. Observing the aircraft well less than 500 ft. off of the wing; traveling in the same direction; Aircraft X took evasive action; slowing the aircraft as slow as practical and requested Aircraft Y state intentions. Aircraft Y then stated (once again) that they had been flying for the last four hours and wanted priority to land. While slowing down; Aircraft X observed Aircraft Y flying straight across their nose to land (once again; well within 500 ft. of the aircraft).In a normal scenario; the aircraft flying the wider pattern should've extended its downwind; allowing for the aircraft on a tighter pattern to land first. Aircraft Y exhibited signs of hazardous attitudes and 'get-there-itis' and wanted all other traffic to accommodate them; even if it meant compromising the safety of other aircraft that had the right-of-way. Speaking from personal experience; ZZZ has many local training operations (parts 61 and 141); rotorcraft (part 61) as well as many transient aircraft operations. While the training operations adhere to FAA-recommended procedures for non-towered operations; many of the transient aircraft do not and make their own procedures out of convenience (such as flying right traffic patterns; entering the patterns on a base; or simply joining the downwind). The sheer frequency of aircraft operating at ZZZ; especially without a present Control Tower; makes operating in the pattern and within four miles of the airport dangerous and seemingly daily pattern conflicts.