B767 pilot reported extreme storm conditions during arrival and final approach caused loss of control and significant airspeed deviations. The flight crew performed three missed approaches and ultimately diverted due to fuel limitations.

Narrative

During preflight in ZZZ1; we were aware that there was a developing severe weather front that was moving toward ZZZ and would impact the airport at some point during our evening. The flight was uneventful until the final portion of our arrival and our hand off to ZZZ Center. We were told that there was severe weather approaching ZZZZZ. We could hear planes requesting deviations to the North of ZZZZZ. Center advised us of the weather near ZZZZZ and asked us to let them know if we needed to deviate. We devoted considerable time and scan to the developing weather and in the cockpit we thought we could stay on our route or perhaps even deviate South instead of North. However; about 20 miles from ZZZZZ Center directed us to turn North. The directed turns and deviations continued until I finally asked why we were being vectored. Center responded that everyone was going that way so he assumed I wanted to follow. The deviations continued to the North and East of ZZZ2 almost to ZZZ3. At that point Center said they thought there was a legal gap and we could make a hard right turn when we were ready back to ZZZ. Again; we did a comprehensive scan with our radar and agreed with Center; and we made the turn. About 15 miles into that turn it became obvious that the route was not wise. We were at 12000 MSL with TAT at 8C. We were not near the freezing level and had plenty of spacing on moderate weather returns. However; we began to experience a great deal of fire on the windscreen followed by at least 3 lightning strikes. We also entered an area of moderate rain and turbulence. The ride deteriorated greatly. I asked Center how everyone else reported the ride through here and learned that I was the first plane they had turned. I asked them not take anyone else on this route because it was not good. We descended into the traffic pattern at ZZZ and they were landing south which seemed odd. They had switched the airport around but the storm front was coming from the South with the best radar picture was to the North at that time. There were reports of rain and strong winds but no shear reports or active thunderstorm on the field. There were definite storms around the field but we worked with Approach to get vectored onto the ILS XXL. We began configuring and got to gear and flaps 25 to attempt to capture LOC and GS. Between 4000 MSL and 1000 MSL the winds were very strong; between 65 and 70 kts. from the southwest. This made the weather a rapidly changing situation. As the airplane attempted to capture the glide path and localizer; I noticed large deviations in the auto throttles and the airplane was above the path and not capturing the LOC. We heard other planes going around and saw a large cell rapidly approaching the final. We decided to break the approach off but both agreed we now needed an immediate 90 to 120 degree turn to the E/NE to avoid the weather that had moved in. I called for the approach to be terminated; we were still 2500 ft. in the air but configured. I did not say go around; but did call for flaps 20; and once we got positive rate gear up. I simultaneously disconnected the autopilot so I could max perform the left turn and we experienced some very rough air. I don't know if it was updrafts and downdrafts but it was very violent. We also experienced massive airspeed changes and it was all I could do fighting the auto-throttles to not overspeed anything. I did not notice until we leveled that I had never asked for clean maneuvering to be selected. The storm/front charged through ZZZ and [we] were vectored for a bit. ZZZ decided to switch the airport and told all of us in the pattern we were going to be vectored to YYL. We had plenty of time to reconfigure and review the approach. We had a good look at the radar from a long final and briefed internally that if we went around we would escape just like the last time with a hard right turn away from the weather. Planes began to land on YYL. We were offered YYR and accepted. Again;no reported wind shear and planes were departing and arriving. We configured and joined the ILS YYR. We were visual and stable at 1000 ft. At 500 ft. I felt the power came way back and we went above the glide slope. I knew we had encountered a gain as simultaneously we got a wind shear caution. We were at 500 ft. so I elected to go around. I called go around; flap 20 and then positive rate gear up. Then it all fell apart. We again encountered either wind shear or just severe up and down drafts. Violent aircraft shaking and rapid auto throttle movement. Our throttles moved almost stop to stop and we got a low speed message at one point while I again disconnected the autopilot so we could execute a rapid 90 degree right turn. We eventually leveled cleaned back up and begin getting vectors. We did report the gain/loss etc. to ATC. We never verified GA illuminated and in hindsight I do not know if I actually got the plane into GA mode. After being vectored around; we were offered the ILS YYL. Planes were again departing and arriving. The last approach was almost identical to the first in that a cell overtook us on final. While approaching the glide slope it became apparent an area of heavy virga was ahead. Much credit to the First Officer (FO); I was focused on getting established on final when he made the case we must discontinue. They said we must break this approach off and you need to turn hard left. Once I looked out I quickly verified what they were seeing. Once again I disconnected the autopilot to effect a harder tighter turn. We experienced many of the same things as we did on approach 1. We began a set of very long vectors and had time to talk with dispatch and ATC. Dispatch changed my alternate to ZZZ4. I told them we had been hit by lightning and I would rather go to ZZZ5 knowing we have maintenance there. They denied this and said ZZZ4. We set a bingo and everyone agreed the worst of the storms were almost through. We reached our bingo fuel and diverted to ZZZ4 without further incident. I learned a lot and think I made several mistakes and wish I had done several things better and different. After seeing the radar feed after the flight it makes sense why we kept getting into the frontal boundary when I was able to see how it was moving across the field. I feel like I got so focused on the weather that I made it my immediate threat which made me not follow procedure correctly on all three go -arounds. Missed [approach] 1 - even though I broke the approach off and disconnected the autopilot to effect a better turn; I needed to execute the missed correctly. I did not methodically step through the steps; especially the key step of bugging clean maneuvering speed at 1000 ft. which would have been immediately. I did not disconnect the auto-throttles which may have been prudent. Since I didn't activate go around the auto-throttles were fighting not only the environment but I still had approach speed bugged. I was so focused on the weather escape I did not execute a discontinued approach correctly. Missed [approach] 2 I believe I reacted quickly and appropriately to the gain. However; once again was focused more on the escape turn than the proper go around. More so; looking back I think it is possible we had Microburst encounter. I was so focused on turning away from the storm that I completely failed to recognize what I believe happened; we went from a gain shear to a massive loss shear. I am embarrassed to say that I lost situational awareness. We should have transitioned quickly and definitively into a wind shear escape maneuver and I did not. Even worse; I chased auto-throttles for speed and once again never got clean maneuvering locked in. It is a good thing we had altitude and a partially clean airplane. Missed [approach] 3 If not for the FO I think I would have possibly flown us right into another massive wind-shift. That discontinued approach was better; but still a bit discombobulated due to the needed hard turn awayfrom the weather. I lost my situational awareness and was overwhelmed by the rapidly changing encounters. I got flustered and the conditions almost exceeded my abilities. I wish I would have executed the missed procedures as advertised and verified. I think I should have converted to a wind shear recovery on missed #2. I wish I would have queried ATC stronger before the turn into the area we got struck by lightning. I would not have turned there if I knew I was the first plane to turn. Last; the long vectors were nice but they are actually too much of a workload. I slowed way down to build time; but a holding pattern that the airplane could just fly why I thought and planned would have been invaluable. I do wish someone in Dispatch or ATC would have just had all of us go hold while the line made it through. However; I think we may have been the only divert that night. In the end; I just think somehow we found the wrong place to be on each approach. The First Officer was outstanding with his inputs and support. That would have been overwhelming without help from the right seat.