C172 Flight Instructor with student reported low voltage indications during cruise. The flight crew diverted and made a precautionary landing.

Narrative

At approximately XA:30 in level cruise flight at 10;000 ft. on an IFR flight; we heard an aural alert and my student noted a Low Volts annunciator on the aircraft's PFD. The Low Volts annunciator extinguished within a few seconds; but both myself and my student noticed the M BUS Amps erratically varying from +24/-10 over the course of a few minutes. Since our destination of ZZZ1 was still roughly an hour away; I instructed my student to request a diversion to ZZZ (which we were abeam at the time). We did not [request priority handling]. The student stated to ATC that we just lost our alternator and we would like to divert to ZZZ.ATC started to vector us; provided a revised clearance; and asked how long it would take for us to get down? Assuming the controller meant to our new assigned altitude of 7;000 ft.; I replied about 3 minutes. Leaving the autopilot engaged; I set a descent rate of 1;000 FPM and began to run through the Low Volts Annunciator Checklist. We cycled the Alternator off; noted that the ALT FLD circuit breaker was still pushed in; and then turned the Alternator switch back on. Engine indications appeared relatively normal for a few minutes while we continued our descent; although the M BUS Volts read closer to 25 Volts. Approach Control provided a few more heading and altitude changes; and cleared us for the ILS XXR as we had some difficulty visually identifying the airport due to the sun reflecting off a shallow haze layer. As we turned onto the LOC; the M BUS Amps dipped again; and we received another Low Volt Annunciator. We were within a couple minutes of landing and had not yet begun to draw off the standby battery. Since landing with main battery power seemed assured; we did not execute the Reduce Electrical Load Checklist. Around this same time; I heard Approach Control mention something to another aircraft about them following an aircraft. I was too focused on monitoring the electrical system; so did not fully grasp that ATC had [requested priority handling] on our behalf; although I do not recall ATC making any direct mention of this or ask for souls/fuel remaining. We continued along the ILS; and were given a change to Tower frequency for landing clearance. As we approached the runway; ATC's prior statement about the priority aircraft became clear to me because there were fire trucks waiting along the runway for our arrival. We had a normal landing on Runway XXR; taxied clear of the runway and to the FBO under our own power and without further incident. In retrospect; I believe the diversion decision was the prudent and safe call to make. However; as this was the first time I've had to deal with an anomaly of this sort in flight; the execution of tasks wasn't totally perfect and there are a few things I would do differently in the future.The timing of requesting the diversion may have been slightly premature as we had not yet run the appropriate troubleshooting checklists. However; given our proximity to the airport at the time and the lack of Towered airports along our remaining flight path; requesting the diversion when we did seemed to make the most sense. Had we run through the checklists first; we would have been further away from our best diversion option. Rather than asking the student to request the diversion; I should have tasked them with running the checklists while I coordinated with ATC. This would have helped me to maintain better situational awareness about ATC's decision to [request priority handling] on our behalf. I could have also more clearly communicated the nature of our diversion; and helped ATC to coordinate the response from the ground personnel more effectively. I believe the decision to leave the autopilot engaged was correct as this reduced the workload of hand-flying the aircraft as we worked to resolve the issue; and increased our ability to exercise CRM (Crew Resource Management). Had we reduced the Electrical Load per the aircraft checklist; we would not havehad autopilot functionality. If we had a longer distance to travel in a future diversion; this would not be something I could rely upon and would need to adhere to the aircraft's checklist more closely.