Light aircraft instructor pilot reported an NMAC with another light aircraft in the pattern at DED airport.

Narrative

We were in the process of a training flight doing pattern work at DED; specifically focusing on normal landings under crosswind conditions. On our final lap of the day is when the incident occurred. We departed Runway 5 for left traffic. Almost immediately after we made our radio call on the CTAF stating we were taking off runway 5; the aircraft behind us in line to depart (Aircraft Y) made their call and announced they were starting to takeoff. From our perspective; it is unclear whether they started to takeoff when we were still on the runway; indicating a possible runway incursion occurred.On their initial departure call; they stated their intention was to make a crosswind departure with no other information given. We proceeded in our climb and made our crosswind turn 300ft below TPA (~800ft MSL). We made our radio call on the CTAF. Shortly after that radio call and turn was made; we noticed the aircraft behind us made their crosswind turn at that time around 300ft MSL and no radio call was made at this point by them.We continued to climb to TPA and begin our left downwind turn; this is when we saw the aircraft to our left in the crosswind below us; climbing. At this point; the aircraft made a radio call 'left crosswind departing 5' with no further information stated on their departure direction or intentions. So at this point; we're approaching TPA in the downwind and the other aircraft is climbing into us from their crosswind. I reached out on CTAF to ask their intentions and if they had us in-sight and to indicate they were about to cut us off in the pattern with their early and low crosswind turn. No response was given. We kept aircraft in visual sight. While they were slightly still ahead of our position; they proceeded to make a downwind turn and they had climbed to approximately 1000ft. This unstated downwind turn is where the loss off separation began to occur. Now we and they were in the downwind at approximately the same altitude. I kept the aircraft in sight off our right and began to make a radio call in CTAF to make contact with the aircraft to obtain their intentions and to inform them we were close. No response from the other aircraft was obtained. They proceeded to drift towards our direction in the downwind at approximately the same altitude (they were approximately ~50ft above us and 100-200ft laterally away) with no response and losing visual contact with the aircraft near; we began a steep descent and right turn to avoid a collision. The crosswind turn to loss of separation occurred very quickly and deteriorated quickly within approximately 20-25s. Contributions to this would be the prior aircraft departing to close behind us; making a crosswind turn behind us and at 300ft; loss of situational awareness for the aircraft behind us; not making accurate radio calls when departing the traffic pattern; changing frequencies when still in the close vicinity of the airport. On our part; I could have recognized the situation deteriorating a bit sooner and taken evasion action in a faster manner to avoid having the other aircraft get exceptionally close to us. This was my first occurrence with a severe loss of separation and will serve as a learning experience should a similar situation arrive in the future. I will also not fully rely on other pilots and aircraft to do exactly what they state they intend to do and remain more vigilant and aware during close proximity to the airport.