C172 Flight Instructor reported the throttle became unresponsive during cruise. Flight Instructor diverted and moved mixture control to cutoff position to reduce airspeed for safe landing.

Narrative

We were on an IFR flight plan from ZZZ-ZZZ1-ZZZ. We went to ZZZ1 with no issues en route and did a single touch-and-go. When we were departing ZZZ; we released the brakes to allow the plane to begin rolling out of the parking space. Person A was on controls and tried to take the throttle fully out; but it was around 1100 rpm. We vocalized the abnormality at the moment; but then the Rpm reduced to normal; we taxiied under 1000 rmp with no issue; and did the run up. During run-up; we were able to bring the plane back to idle after mag checks with no issues. We proceeded with the flight; and after the touch-and-go at ZZZ1; we got back on with ZZZ Approach and reopened IFR back to ZZZ. We were told to climb to 6000 and were direct ZZZZZ. Person A needed a break after flying; so I took controls to allow her a moment to rest and get a sip of water. I held the heading and was on the climb to 6000. Initially; as we got to 6000; I accidentally went past to just over 6100; I immediately noticed and reduced throttle and descended back to 6000 to correct. When we stabilized at cruise; I asked Person A if she was ready to come back; and passed her controls; advising we run the cruise checklist. When we went to run cruise; I noticed the rising airspeed. I told her to watch the airspeed; expecting her to correct. She made an attempt but did not vocalize what she was doing. I reiterated to slow the airspeed and advised to remove some power. That is when she vocalized that the throttle control was fully out. I immediately took controls and began trying to move the throttle handle full range of motion. I quickly determined that she was correct; and the throttle control no longer had any response with the engine. The RPM were around 2600; and airspeed was rising well over 110. I advised ATC we were having an issue with our throttle and were trying to resolve it; asking to deviate from 6000. We began to oscillate on altitude; allowing the airspeed to slow on each climb. We were attempting to decide what to do and told ATC that we possibly would be requesting priority handling and would need a diversion. They advised we were 15 miles west of ZZZ2; so we began heading that way. Person A began briefing the airport; and when we saw the runway length of XXL; we decided it would be suitable. I requested she double-check that there is not a checklist for this; and that we review the landing distance data we had calculated prior to the flight. While we were orbiting trying to resolve; we were in and out of clouds. At one point inside of a cloud; our RPM exceeded 2800 and our airspeed was over 130 kias. Person B was in the back seat as an observer on the flight. He was coming along to see the airspace and observe as a new CFI. I asked him if he had cell service; which he confirmed; and asked him to contact Person C. Person C made the suggestion to use the mixture as throttle. Initially; each of us had planned that we were going to need to cut the mixture and descend power off; gliding to the runway to remain in control of airspeed. We were trying to determine what distance from ZZZ2 would be most appropriate to cut the mixture. With Person C's suggestion; we were luckily able to reduce the RPM. I found that there was a very fine line of pulling too much mixture that made the engine sputter; and just enough that kept it running. From my recollection; we were able to reduce RPM just enough to descend at a safe speed; but RPMs were still high. When we came out of the clouds around 2500; the airport was right in front of us. I realized how high we were; and that we would definitely make it gliding; so I cut the mixture. We did a forward slip and came down over the threshold; going around 112 kias with no flaps. It was gusty from the left; and I was focused on holding centerline. I had Person A reading me the airspeeds out loud as we slowed down; and the last speed I remember her saying was 75 before we settled down to the runway. We exited XXL at [taxiway] 1and came to a complete stop. The airport ops manager and fire chief met us; and coordinated to have the FBO tow us to parking.