B777 flight crew reported a near miss during taxi with an aircraft that was pushed back directly into the path of the B777.

Narrative

The occurrence happened as we were preparing to taxi from parking position X. I was seated in the observer seat; with the Captain as the Pilot Flying (PF) and the First Officer as the Pilot Monitoring (PM). The taxi out from the parking spot only required forward taxiing; so no pushback was needed. After receiving ATC taxi clearance (Taxi to Runway XX via 1; 2; and 3); the Captain and FO cleared the area; and the PF started to add power and move slowly to turn left and join 4. I also conducted a quick scan before we started moving; and as the aircraft began taxiing; I briefly looked down at my EFB to review the taxi chart; and suddenly; I felt the aircraft abruptly stop.When I looked up; I saw the Captain pointing towards a Boeing 737 that was being pushed back directly into our taxi path. The 737 was being pushed at a high speed into the taxi lane we were about to use. This is a non-movement area; so ATC was not responsible for controlling the pushback. However; it appeared that the pushback crew failed to properly check the area before initiating movement. From my perspective; the 737's pushback seemed rushed; as if they were attempting to move ahead of us. Fortunately; the PF reacted quickly and stopped the aircraft in time; preventing a potential ground collision.This is not the first time I have observed fast pushbacks in this area without proper situational awareness. Recommendations: Ground handling crews should be reminded to conduct thorough visual checks before initiating pushback; especially in non-movement areas where ATC does not provide oversight.Coordination between pushback staff; flight crew; and other traffic should be improved. In non-movement areas; a radio call from an aircraft about to start pushback should be encouraged.