An air carrier Captain reported the tug driver suddenly stopped their push to avoid an aircraft taxiing on the ramp.

Narrative

Near collision on ZZZ ramp while pushing from Gate 1 due to high-speed Aircraft Y taxi cleared around the corner to same spot my crew was cleared and pushing to. Contributing factors likely included inadequate wing walker monitoring of ramp state and or communication with push tug driver; flight crew lack of situational awareness via ramp radio calls; and flight crew complacency accepting a conditional push clearance. Submitting a report in parallel with tug driver/push Captain who stated they intended to submit a report; stating I can't tell you how close that was. I've been on this ramp for XX years and it's crazy who they're hiring upstairs as ramp controllers pretty much off the street." When we originally called for push; ZZZ ramp cleared us to push to Spot 2 after Aircraft Y vacates and taxis with Ground. I energized beacon; released parking brake; confirmed steering off; and communicated to push crew via voice and hand signals. They confirmed with voice. Several minute wait prior to Aircraft Y taxi. As Aircraft Y began to move my push crew called up and announced 2 was clear and they were commencing the push. After approximately 30 to 40 ft. of pushback we suddenly stopped. My First Officer (FO) looked out the window and said a plane was taxiing high speed and came to a sudden stop right before where we were pushing to. Simultaneously the tug driver communicated they were stopping the push due to traffic behind us in the alley. I ordered them to tow us fully back into the parking spot; where I reset the parking brake; turned off the beacon; and announced to ramp the dangerous conflict and my actions; stating that I would not push again until they confirmed all aircraft were clear and we were cleared for immediate push. Ramp apologized multiple times; profusely; for forgetting we had already been cleared to that spot. Cause: Primary cause was Ramp Controller clearing Aircraft Y to taxi to the same spot we were cleared to push to. Secondary causes include conditional push clearance; delay push until Aircraft Y vacates that spot; wing walkers presumably inattentive and not communicating presence of Aircraft Y until tug driver saw the conflict; and Aircraft Y taxi speed. Conversation with tug driver indicated a recently increased problem with ZZZ Ramp Controller training and professionalism due to post COVID hiring and training. As a Captain I've noticed a degradation as well but unable to place it in historical context or understand the difference between; e.g. ZZZ Ramp Controllers and ZZZ1 Ramp Controller staffing. Note that I realize this isn't ATC but the report has no boxes for ramp controllers and other ground handling incidents. Suggestion: Recommend reviewing video/audio of this push and maybe a week of ZZZ ramp pushes to assess alleyway dangers and need for increased training/supervision of ramp controllers and push crews. At a minimum ZZZ ramp safety stand down and review of this situation. I will personally no longer accept conditional push clearances on the ZZZ ramp because it's clear the rest of our system doesn't have my back; literally. ZZZ1/ZZZ2 should consider whether conditional push clearances are appropriate operating procedure especially in non-dead end alleyways."