PA28 Flight Instructor with student reported gusty winds caused a loss of control during landing resulting in a runway excursion.

Narrative

I was flying with my student; we were doing an instrument training flight to ZZZ1. As we took off out of ZZZ at around XA00 winds were a bit shifty as we got over the trees; but nothing that we weren't used to. We were flying at 3000ft over to ZZZ1 weather was good nothing out of the ordinary. As we got closer to the field we listened to the weather. It was reporting gusty and winds were favoring runway XX. About 15 miles out from ZZZ1 I called ZZZ1 tower and requested a touch and go. They told us to report midfield right downwind for XX. We started descending to TPA. Soon we were midfield right downwind and we reported. The controller cleared us to land. My student was in control. As we were in the downwind I briefed my student that it could be a bit gusty and to prepare by coming in slightly faster to account for the gust factor. He understood and we continued the approach. As we descended he had an extremely stable approach that would've placed us right on the 1000 footers. I told him that although we were slightly fast; we had plenty of room to touchdown on the runway and it didn't matter if we missed our aiming point as long as it was safe. My student was correcting for the crosswind. As my student was rounding out there was a huge shift in wind which pushed us down and to the right. It happened fast and I took control. At this point we were headed for the grass. I did my best to get us to the center of the runway while simultaneously going full power to execute a go around. I'm not sure if the controller noticed or not because he didn't ask if we were okay. The controller proceeded by asking our next intentions. I told him we were going back to ZZZ. After getting back in the air my student asked what happened. I expressed to my student that it felt like wind shear. His approach was stable; airspeed and configuration were all satisfactory. I then asked the controller for their information so I could contact him and talk about it once I landed. I contacted our director of operations; and expressed that I wanted to call him once I landed. He asked if I was okay and I said yes. Heading back to ZZZ I asked my student if he still wanted to fly and he said yes. I told him because of what I thought was wind shear; I would take the landing back into ZZZ. He agreed. On the way back I listened to ZZZ2's weather and sure enough it now was reporting low level wind shear. This confirmed my suspicion. As we approached ZZZ my student ran his approach checklist and started descending. We were slightly above TPA when we started feeling significant changes in wind velocity and direction. I took controls. The winds on ZZZ2's ATIS were favoring [Runway] XY. We were looking for the runway and ended up approaching the end of XY. I wanted to get a feel for what the winds were doing so instead of going straight in; I flew straight and turned left into the left downwind for XY to fly the pattern. As I was flying we noticed moments where we had a sudden loss in lift. And strong shifting crosswinds. I knew that it was going to be a difficult approach.As we were on final for XY; it felt like we had a strong tail wind pushing us; which I thought was abnormal since the ATIS was saying it should be a headwind. I know that because we were using ZZZ2's weather service; since ZZZ does not report weather. I decided to fly over the field to verify the wind sock. I asked my student to look out and see what direction it was favoring. Both of us verified that it was a direct headwind for XY. After verifying; we turned left downwind for XY once again. This lap the winds felt a lot calmer. The winds still felt like a tailwind; but I didn't experience as much turbulence as I did in the first lap. On final; the approach was stable. I prepared my student and told him to get on brakes; considering I do not have toe brakes. My student confirmed he would be on the toe brakes once we landed. As we were approaching; everything looked stable. As we descended over the trees; I pulled power for the short field landing. Before I was able to round out; we experienced a significant loss of lift and impacted the runway with force. We impacted and bounced. Then we impacted again and I tried to control the plane to maintain centerline. At some point during the porpoise; we lost directional control and began skidding left. Eventually; we entered the grass left of the runway and came to a complete stop. I shut the plane off and my student and I exited the aircraft. There was no injury. I knew that the plane was damaged due to the force of the first impact. Additionally; I saw the prop bent when we went into the grass. After exiting the aircraft; I proceeded to call the Director of Operations.