Air carrier pilot reported the aircraft rolled uncommanded during marshalling when the brakes were released before chocks were placed. Pilot applied brakes to stop aircraft.

Narrative

Operating flight ABCB from ZZZ to ZZZ1 on Day 0 date. It was an abrupt pushback initially and also at the end of the push. Once we had started engines I could tell they were struggling with the tug and tow bar. The mechanic asked me to go ahead and set the brake. I asked him if they were having issues down there with the tug and/or towbar. He said that both shear pins had broke on the abrupt pushback. Apparently the tug they used also has issues with throttle/gas position that requires gunning it versus easing the gas to move not as abruptly. At any rate; I told the mechanic to standby in headset while I called dispatch+AMOC (Alternate Means of Compliance) due to the broken tow bar needing to be written in the logbook. We had already completed the after start checklist so I asked the FO to re-start the APU. Once that was started I shut down the left engine to conserve gas for the block turn back. Once I got dispatch on the line; she connected maintenance control to us. I let AMOC know that we had sheared the pin on the tow bar and would need to put it in the logbook and make sure they were good with a block turnback. AMOC agreed and after I hung up with them plus dispatch I began coordinating with the mechanic on the headset for the turnback. He asked if I could make the sharp turn back into parking (due to the way they had pushed us back on the taxi line) and I indicated I couldn't. I then asked him if we could get towed back in and he said they only had one more tow bar and didn't want to break it. So he asked if we could taxi around the fuel farm and back around to park at Spot X. I indicated that we would do that. So I released the mechanic off headset and proceeded to taxi on one engine around the fuel farm and back to spot X. While taxiing aloud the fuel farm we completed after landing flows. Once we pulled into Spot X I was marshalled to a complete stop. I set the parking brake and called for the shutdown checklist. The lead marshaller with orange wands was either being given training or was training the person behind her; but something about the way she was doing her job did not instill confidence. I made a comment to the FO that the irregular nature of the block turnback isn't probably the best time to do training for ground ops. I made the comment because after we parked and set the brake; the lead marshaller just kept the X up for a long time. And I was surprised that we hadn't been given the chalks signal after a few minutes. Finally the marshaller signaled that stairs were up. I then immediately gave her the chalks signal back (instead of acknowledging the stairs signal with a thumbs up) with a hands up gesture after to indicate are the wheels chalked? She then looked around; yelled at someone around the airplane and then gave me the chalks in signal back. I said to the FO at that moment that I didn't trust that was the case and that I was going to outside and verify. The jumpseater; a captain; had opened the main entry door already and the load/ramp supervisor was on the way up the stairs. I asked the jumpseater if he would verify that the chalks were in since he was closer to the entry door. He stepped out on the stairs and asked the ramp supervisor coming up the stairs if the airplane was chalked. Apparently the ramp supervisor then asked someone else who said yes. (This last sentence I didn't hear personally because I was still in my seat. This is what was told to me by the jumpseater and the ramp supervisor after the incident) The jumpseater said back to me that chalks were in so I then looked down and release the brake. While I was looking down my FO yelled out we are moving or I think we are moving. I then looked up; felt the sensation of movement and caught it out of the corner of my eye and slammed on the pedal toe brakes and immediately reset the parking brake. I guess at this same moment the ramp supervisor was trying to step inside the main entry door and managed to get in the airplane as it was rolling. All of this probable happened in a matter of 3 seconds. It wasn't until I looked back towards the main entry door that I realized how much we had rolled back. Probably about 5 feet. The stairs were completely misaligned with the door with at least 50% of the door opening no stairs; just straight down to the tarmac below. I then directed everyone into the main seating area of the cockpit and closed the main entry door. I said that nothing else would be done until the aircraft was properly chocked and the stairs reset in place. Once that had occurred I started asking the ramp supervisor in the cockpit was exactly had just transpired and I could tell he was visibly shaken; possibly because he could have been close to falling out the door or off the stairs. I don't know that for sure; but he definitely looked shaken. I then went down the stairs where the ramp crew had gathered and began having discussions with several people. Initially nobody was coming forward as being in charge or the person responsible for the safety of the ramp crew. I talked to the lead marshaller and asked her why she gave me the chocks signal if they weren't in to which she said she has asked someone else. From my brief interactions with the ramp crew I didn't pick up on anyone with safety accountability that was willing to explain what exactly had gone wrong with the signals being given. I ended the conversations by saying that it was a very serious thing that happened and we were lucky no one was injured. After I returned to the cockpit the gateway manager came up and said that he would be getting to the bottom of what happened. That was essentially the last interaction that I had with anyone concerning what had transpired. We then shifted gears to again getting the airplane safely to ZZZ1; which we did. I'm very glad that no one was injured during this event but I believe that was pure luck. Some of the ground ops people just didn't seem to grasp how serious of an event this was and how much worse it could have been.