Flight instructor with student reported an NMAC at a non-towered airport with an aircraft that flew close behind them then flew below them while on final approach.

Narrative

While flying a practice ILS approach a faster airplane from behind; also on the approach; passed underneath us. The aircraft I was instructing in; Aircraft X; was established on the ILS 30 at LVN talking with other aircraft in the traffic pattern when it was noted a faster moving aircraft (Aircraft Y) appeared also on the same approach. Radio calls were used to coordinate a low approach at the airport and I asked if the other aircraft on approach had us insight and advised we were slow moving. They stated they were already in front of us; which was incorrect and this was pointed out on the CTAF. Foreflight gave a warning of less than a 0.25 nm mile and less than 500ft; and I asked if that airplane was going to fly underneath us as I began instructing my student to add power and stop her descent. Looking at ADS-B data it appeared they were 400ft underneath us as they passed and my plane turned 90 degrees off course and departed the area climbing away. No standard communication was made from that pilot inbound to alert others on frequency to their intentions; until it became dangerous. Key factors are hazardous attitudes; poor radio communication; and dangerous flying.