A321 Captain reported the aircraft rolled forward a short distance at the gate when the tug failed to hold the aircraft steady after engine start.

Narrative

Due to MEL we had no APU and had to start Engine 1 at the gate. After engine start and completion of External Air Start procedures we proceeded to pushback. When I released the brakes we started rolling forward pushing the tug towards the terminal. I stepped on the brakes as the driver was quick to say set brakes several times. Meanwhile while I was stepping on brakes the FO was saying you are moving as well. I stopped us. The driver said to please try again. Part of the issue was that the jet bridge was moving away while we were rolling; leading to a momentary question about who was moving. We only rolled about a foot or two but it seemed like more. I asked for the wing walkers to make sure we were clear and then we tried again; and again we pushed the tug. He asked me to try a third time and I said I wanted another tug. Finally we got a supervisor to drive the tug. Initially I pushed him as well; but since I was quick to the brakes each of these times we hadn't moved much. Finally the driver was ready to push at the same time I released the brakes and he pushed us back with no further issues.Cause: I believe the tug they were using has a problem with the parking brake. It feels a little downhill at that gate so with the engine running at idle on the NEO the brakes or the weight of the tug would have to be significant to prevent movement. Tug driver did a great job yelling for brakes. FO did a great job recognizing our movement. Supervisor did a great job getting us out without further delay. That tug needs to be inspected.