Captain reported as ground conflict resulting in evasive action.

Narrative

Pushed back normally off gate on taxi line. Once after start checklist was completed FO called ramp for taxi. Ramp instructed us to follow Aircraft Y aircraft passing on our right on 11 taxi line then follow them to spot X1 and contact ground. In the mean time aircraft Z had pushed off a gate further east approximately at gate Y1; so we would have to move over to 22 taxiway to pass aircraft Z; then back to 12 to exit spot X1 as directed by ramp. Shortly after ramp amended our instructions to follow aircraft A passing on our right. That aircraft took maybe 2 minutes to pass us. When aircraft X was past we began a single engine taxi and I began a right turn to move us over to 22 [Taxiway] behind aircraft Y as instructed. We did clear left and right prior to beginning the taxi. As we started the turn the FO authoritatively said stop. I immediately stopped the aircraft. Just then another commercial aircraft passed us from behind taxiing east on 22 [Taxiway] at a rapid speed. There was no updated instructions from ramp tower and we could not initially see the commercial aircraft due to facing east. We received a call from the flight attendant station that one of the flight attendants had fallen due to the rapid brake application but was okay to proceed. I notified ramp tower what happened and that either they didn't update our taxi instructions and/or aircraft Y didn't give way and was taxiing at too high a rate of speed for the ramp.