Air carrier Captain reported following taxi clearance; noticed an aircraft being pushed back in close proximity resulting in heavy braking and Flight Attendant injury.

Narrative

Cancun possible taxiway incursion and FA injuryAfter pushback from gate; we were issued taxi instructions to taxi to 12L via B-C follow Aircraft X. First Officer (FO) acknowledged the instructions and I read them back to him. We began a slow taxi as we elected to single engine taxi to the runway. Aircraft X we were told to follow was already well ahead of us. As we passed C7 the FO started the number one engine. FO stated the delayed engine start was complete. I turned right on E3 and continued to follow Aircraft X. As I made the turn I noticed an Aircraft Y pushing between us and Aircraft X we were following (maybe gate 54). Aircraft Y appeared to be stopping as we approached but then appeared to continue to push. It looked like there could have been a possibility of us being in their way and wasn't sure if the push crew could see us. None of Aircraft Y pushcrew seemed to be looking at us. At the same time ground called us and told us we were supposed to stop at E1 and call the other west ground to continue. I can assure you that I never heard this clearance given and neither did FO. This happened at the same time as I became very uncomfortable with the proximity of Aircraft Y. I applied firm brakes and stopped the airplane suddenly. I am not sure how fast I was going but believe I was going less than 10 kts. The East Ground Controller told us to contact the West Ground. We did so. The new Ground Controller issued us instructions to continue if we could maintain clearance. The wing walker from Aircraft Y assisted us by giving a thumbs up as we proceeded past Aircraft Y. The FO also mentioned that our wingtips were well clear of conflict. I called the flight attendants and asked if everybody was ok. The Flight Attendant [FA] who answered said they were seated already for takeoff and ok. We took off shortly after that. About 15 minutes into the flight; I called the flight attendants and mentioned that it was all clear to get up as I briefed them to stay seated until I called due to the weather in the area. The lead [FA] called right back and said that when I abruptly stopped the airplane; FA was still up and securing the galley during the taxi and bumped her head and back. The lead FA said FA was just going to stay seated for a while as she was not feeling well. The lead FA also mentioned that FA took some [medication]. I queried the lead FA if she or FA felt a need for MedLink. She assured me that was not necessary. After a few minutes the lead FA called again and told me that FA got up and continued her duties. I spoke with FA later in the flight and she tried to assure me that she was ok but needless to say I feel completely responsible and terrible about the situation. I did not intend to stop the aircraft so quickly but did feel compelled to cause our aircraft to hit each other. FA [and] the lead FA continued to assure me that she was fine and had no intention of reporting the issue.