Air carrier Captain reported uncommanded aircraft movement at the gate prior to push back. Push crew replaced the tug and continued the push back.

Narrative

After break release and just prior to push; cargo door on aircraft opened. I attempted to contact tug driver to determine what was going on. I could not get ahold of the tug driver. The cargo door closed. At the same time the aircraft started to move due to the wind. I finally got ahold of the tug driver. I asked why I couldn't get in touch with her. She explained that she had been off the headset talking to supervisor. I explained that they should maintain contact with me once brakes are released. She then noticed that the aircraft had moved and asked me to set breaks. They had to remove and reattach tug because of the aircraft movement. At this point we proceeded with pushback. Pushback was stopped again because the wind moved the tug. Was asked to set breaks. Tug driver said they heard a noise and wanted maintenance to come out. Ground maintenance came out and inspected tug. Aircraft maintenance came out and inspected nose gear. Asked aircraft maintenance if they needed a write up and they said they would take care of it. At this point ramp personnel got a bigger tug. We continued the push back without further problem with the bigger tug.