GA Flight Instructor conducting a Visual Approach reported an NMAC with another aircraft not communicating with Tower.

Narrative

While conducting a visual approach into EFD on a dual training flight; shortly after contacting tower; on approximately a 270 heading; an aircraft crossed about a half mile in front of us prior to turning to an east heading on a head-on collision course (Aircraft Y) at 1500' MSL. Tower issued us a traffic advisory. Once observed; we began a turn to the right. The traffic then turned to the left and appeared to continue on a south heading away from us. We continued a turn back to the west and descended to 1100 feet to enter EFD airspace and give a buffer should traffic turn towards us. The traffic then turned in a 180-degree turn to the north. At this point; on board ADS-B showed traffic about 600' above on course to cross behind our flight path. Both my student and I had traffic in sight. Once over top of us; traffic began a sharp left turn and a descent to only 200' above; according to our onboard ADS-B info and 400' from ADS-B track logs. Due to the unpredictability; I deemed our best action to be a quick descent to 900 feet and maintain a west heading. Traffic briefly tailed us at 200 feet above before leveling off on a south; southeast heading. EFD tower handled the situation to the best of their ability and gave us as much info as possible; given the quick deterioration of the situation.