A flight instructor taxiing at a towered airport reported a ground near miss with another taxiing aircraft after the tower ceased operations.

Narrative

The event occurred on Taxiway X at ZZZ airport. Aircraft X has landed Runway X and cleared the runway on Y; following a radio call announcement. In the meantime; Aircraft Y announced short final; then landed Runway X and continued deceleration to exit on Taxiway Z. Aircraft X has initiated movement and announced crossing Runway XX; turned landing; taxi; position and strobe lights on for crossing the runway. In the meantime; while Aircraft X was crossing the runway; Aircraft Y has announced exiting Runway X on Z and intention to move to Runway X via X. Aircraft Y has proceeded to X from X/Z intersection; while Aircraft X was approaching from the left of Aircraft Y as it was clearing Runway XX.Two aircraft have missed each other within a few feet and evasive action had to be taken by Aircraft X. As an operator of Aircraft X on an instruction flight; I was aware of the Aircraft Y on Z; and was expecting that the Aircraft Y would give way to the Aircraft X. After listening to the radio recording; the two radio calls happened within 15 seconds of each other. As an instructor; I missed to correct my student's radio call 'ZZZ Traffic; Aircraft X crossing XX on Y; ZZZ Traffic' - the aircraft was crossing XX on X; Y is a taxiway that is parallel with XX; where the student exited Runway X. The student could have also announced the intention to taxi to Runway X to make the radio call more complete. Considering that the Aircraft Y's radio call was followed shortly after the Aircraft X's radio call; and that we had all our lights on and were visible to the Aircraft Y; we assumed the Aircraft Y intended to taxi after giving way to the Aircraft X. However; Aircraft Y has initiated movement; without checking that the taxiway was clear to the left of its position.Moving forward; assuming someone will give way after announcing their intentions is not the safest approach; and we could have clarified and should clarify next time over the comms. Additionally; more clear and correct radio calls could have improved the safety of this situation. So was debriefed with the student and I will pay closer attention to the radio calls made by the student next time and enforce the usage of taxiway diagrams in both familiar and unfamiliar airports at ALL times. I believe that additionally; more regulatory guidance should be given to pilots in relation to who has the right of way in an untowered field; which should not substitute clear communication and the principle of seeing and avoiding.