Air carrier ground personnel reported towbar detached from the aircraft during pushback resulting in the aircraft rolling uncommanded before the flight crew set brakes.

Narrative

Towbar came detached from nose wheel of aircraft as push was initiated due to improper attachment to aircraft. aircraft moved approximately 3 inches back before disconnect and promptly stopped. Brakes were reset and scene was frozen in order to have aircraft and towbar inspected. This flight was also originally a quick turn and during the time at the gate it was determined that this aircraft was going to need an airstart. As a result; I had to go and find a new pushback (towbar) and send my crew to go find an airstart machine. Once the airstart machine was brought back to the gate by my crew; I attempted to turn it on and discovered that it would not. I then had to go locate a new airstart for and there was not one anywhere else on the northeast side. Since you cannot tow airstart units through breezeways; I did not look on the south side of the concourse. We also had a very large 400 pound wheelchair show up around [gate] X which took 3 people almost 10 minutes to load. Finally; this was my first shift out of lead training on my base schedule.