GA flight instructor reported pattern entry as another aircraft did also resulted in evasive action and an NMAC.

Narrative

My student and I were heading back to ZZZ; an uncontrolled field; after a landing lesson at ZZZ1. The traffic pattern was very busy so we entered on an 8 mile; 45 to the downwind of Runway XX. We made all of our radio calls at 8 miles; 5 miles; and 3 miles. At 3 miles; we hear that another plane has entered the traffic pattern on a 3 mile 45 degree entry to the downwind. We immediately descended below traffic pattern altitude because we didn't have them in sight; and we asked them if they had us in sight; in which they said they didn't. They end up doing a left 360 degree turn to re-enter the traffic pattern. I feel that if we hadn't descended below traffic pattern altitude; they would have been within approximately 200-300 feet horizontally to us at the same altitude. Transient aircraft who come into ZZZ and are unfamiliar with our traffic pattern have trouble entering it in a safe way while it is busy with training aircraft. A lot of times; transient aircraft will enter directly on a downwind or a base leg; instead of entering on the preferred 45 degree entry. This cause a lot of traffic pattern conflict at ZZZ.