Flight Instructor with student reported that after landing and stopping on the runway another aircraft executed a missed approach overhead and close to the Instructor's aircraft.

Narrative

During a training flight me and my student were in the traffic pattern at MMV in left traffic for Runway 22. There were multiple helicopters in the pattern for the taxiway parallel to the runway as well and Aircraft Y on the Localizer approach to Runway 22. My student and I were in the downwind configuring and preparing to turn base when Aircraft Y called he was 5nm out for a low approach on the Localizer practice approach. I confirmed his distance on our ADS-B receiving MFD and chose to complete the traffic pattern as he showed to still be 5 miles out. We called our base turn and turned base. I continued to evaluate his position through out base segment. Upon turning final he had closed the gap from 5 nm to approximately 3nm between us. I continued to verify his position as we flew our final approach in and continued to report our position while stating we planned to. do a touch and go and stay in the pattern. During the final 1/4 mile of our approach we were receiving Traffic alerts from the G1000 as he was at same altitude and less than a mile. Upon landing I told the student to full stop as I was concerned if his ADSB was just wrong or if he was really that low and flying at us. I then spotted Aircraft Y approximately 70-100 feet above us while we were on the runway and of to the side about 30-40 feet. Aircraft Y's last call was a executing a missed approach departing to the north which came at around a 1/4nm final for Aircraft Y. Being a CFII I had expected them to be approximately 500 feet AGL at least as the minimums for this approach were 660 MSL or 440 AGL. My first assumption upon reflecting on this occurrence was that they meant they were on the ILS approach not the LOC however they would still have been too low for that approach as the minimums are at 200 AGL and missed would've started before the runway causing them to be well above 200 AGL by the time they passed above us.