Flight Instructor reported an NMAC in cruise requiring evasive action to avoid a collision.

Narrative

We were departing ZZZ via the crosswind departure. Upon flying out towards ZZZ1 ATC alerted company traffic ahead not talking to ZZZ same altitude. About 4NM to our east. ATC advised to change frequency early. Aircraft Y seemed to be flying the same departure; significantly east of the departure. We ended up level with them heading the same direction. We both made calls to the SE practice area. indicating a similar direction we climbed on course keeping adequate space; maintaining Aircraft Y in sight visually. We had noticed Aircraft Y creeping toward us on ADS-B; nothing egregious. We sidestepped our intended path to give them more room. Once 12NM from ZZZ [VOR] we switched to making a call to the stack to make preparations for practicing approaches at ZZZ2. At this time both us(Aircraft X) and Aircraft Y were flying at 5500. My student and I noticed that Aircraft Y started to make a turn directly into us without climbing or descending. IP (Instructor Pilot) took controls and performed an emergency descent while Aircraft Y maintained altitude shortly and started to climb after we started a significant emergency descent. While IP believes only reached 152KIAS; plane was squawked as precaution to inspect as negative G was induced and flight data was not readily available. IP and SP (Student Pilot) worked to maintain a visual on Aircraft Y during the descent. Suggestions: Visual scanning techniques to be reviewed with students. Students will monitor ZZZ frequency when (assumed) transitioning from the downwind departure to the crosswind departure.