Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Situational Awareness; Confusion
Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence; Inflight Event / Encounter CFTT / CFIT; Inflight Event / Encounter Unstabilized Approach
Air Traffic Control Issued Advisory / Alert; Flight Crew FLC complied w / Automation / Advisory; Flight Crew Regained Aircraft Control
Software and Automation; Weather; Human Factors
We were on approach to RWY 30 into BZN on the RNAV RNP 30 from Livingston transition. We were getting moderate turbulence with the wind around 600-70 kts below 10;000 [feet]. We had thoroughly briefed the approach prior to the descent. While on the approach around 5;000ft Tower advised us the an aircraft had circled to 30 from ILS 12 and during the approach experienced a 2;000 fpm updraft and moderate turbulence on approach but braking action was good upon landing. Around LAVPE; the Captain called for below the line landing checklist while we were descending on the VPATH. I ran the checklist and then looked at my Jepp Approach Chart to confirm what the missed approach altitude was; it was right after that we got a 'Caution Terrain' aural warning followed by Tower telling us we had a low altitude warning. We then got another caution terrain warning and the Captain disconnected the AP and started to climb; I saw at that point we were below the VPATH somehow. We started to climb back towards the path and then got the RWY in sight and descended for a normal landing. Some time between me going heads down to run the below the line landing checklist and MOSNE intersection we got low and I still do not know how. We were on AP and VPATH just a few seconds prior; I don't know if we got a downdraft or what but I'm still confused as to how the aircraft got low. We decided to continue to the approach instead of going around and I believe that was the safer of the two options giving the wind gusts and moderate turbulence in mountainous and unfamiliar (for me first time to BZN) terrain.We both had a long day (I commuted in that morning because I missed my last commute the night prior due to operational delays) and had been fighting turbulence all day long on ZZZ-ZZZ1-ZZZ-BZN routes which probably made us more tired than normal. This was my first time into BZN and the low visibility; moderate icing from the snow storm; and moderate (or greater) turbulence on approach made it tougher to concentrate and multi task.I'm not sure how we got below the path on approach as I was Pilot Monitoring and heads down running the checklist so I have no suggestions as of now. I should have done a better job monitoring our descent and VPATH even while conducting the Landing Checklist.
B737 First Officer reported distraction prevented noticing Pilot Flying descended below path; resulted in ATC low altitude alert and GPWS terrain alert.