Always,
Ravina and Rav Ashi organized the Bavli. Ravina died a good number of years after Rav Ashi. There was some minor editing afterward (what we call the Rabban Savaroi). There are some academics that argue for more extensive editing after Ravina and Rav Ashi but that isn’t based on any science. It’s not something they “found.” It is based on rayos from Shas (and is not very persuasive).
To answer the OP. The theory that made the most sense to me is that there was an earlier proto-Gemara that was composed (but not written) with specific language to remember. I’ve seen it argued that Abaya and Rava composed the proto-Gemara. But every scholar in every town had comments and a mesora on every aspect of the Mishna and the proto-Gemara. Rav Ashi and Ravina gathered as many chachomim as they could to create a final version of explanations and comments on the Mishna. They edited these together to form Talmud Bavli.
The generations immediately after them wrote down the Talmud Bavli (along with some further edits) and shipped copies to Jewish communities far and wide.