GA Flight Instructor with student reported an NMAC while in the pattern of a non-towered airport. Instructor took evasive action and continued to a landing.

Narrative

Training flight with my student in Aircraft X. Doing pattern work; making proper radio calls and position reports. I was aware of another local aircraft that was flying all over the traffic pattern area erratically; but who was not actually flying the traffic pattern. He crossed midfield below TPA (Traffic Pattern Altitude) while I was midfield on the downwind. I got visual as well and confirmed on ADS-B. We began our descent and I asked if he had us in sight. He said he did not. I told him where we were in relation to him and that we were beginning our base turn about 1 mile ahead of him; same altitude. I had to quickly take evasive actions as he began to descend into us; and he finally corrected his location after I queried him for his intentions. We corrected course; turned final; and executed a full stop landing. He then circled around after stating he was intent on landing; entered the upwind for the same runway (17); and crossed midfield at 400 feet AGL to reenter the downwind leg. He would have run right into me while flying a non-standard traffic pattern and exercising poor communication if I had not taken evasive actions and alerted him to the danger he put us in.