Flight Instructor in a C172 reported a NMAC with a faster aircraft which overtook them during their initial climbs from a non-towered airport.

Narrative

This incident occured at approximately at XA:10. My student and I had just departed Runway X and were climbing for a left crosswind departure to ZZZ1 Airport. Only a few moments after departure; the other aircraft took the runway and began their departure. The other aircraft appeared to be a warbird or trainer and appeared to have much better climbing performance. My student and I were making calls such as initial departure; departure leg; and left crosswind departure to the northwest. The other aircraft was not making position reports. Though radio communication is not necessary at a pilot controlled airport; I feel it is relevant to the incident. Due to the discrepancy in the performance of our aircraft; the warbird quickly caught up to my aircraft. We were tracking the other pilot's position on ADSB-in/Foreflight so we could see he was quickly approaching. At the closest point; the other aircraft was approximately our same altitude and 100 to 150 ft. away laterally upon which; the pilot peeled out to avoid collision. It is unclear if they ever saw us at all. It should be noted that ZZZ is particularly busy on this day for their weekly fly in. I believe the pilot of the other aircraft was trying to depart ahead of another aircraft on a base to final turn to avoid a departure delay. It is the opinion of this pilot that the pilot of the other aircraft created an unsafe situation. Shortly after the incident; I contacted FSS on the radio to make a report and was told to file this report.