COVID-19 pandemic has brought unprecedented challenges to the world, and vaccination has been a key strategy to combat the disease. Since Twitter is one of the most widely used public microblogging platforms, researchers have analysed COVID-19 vaccines and vaccination Twitter discourse to explore the conversational dynamics around the topic. While contributing to the crisis informatics literature, we curate a large-scale geotagged Twitter dataset, GeoCovaxTweets Extended, and explore the discourse through multiple spatiotemporal analyses. This dataset covers a longer time span of 38 months, from the announcement of the first vaccine to the availability of booster doses. Results show that 43.4% of the collected tweets, although containing phrases and keywords related to vaccines and vaccinations, were unrelated to the COVID-19 context. In total, 23.1% of the discussions on vaccines and vaccinations were classified as Pro, 16% as Hesitant, 11.4% as Anti, and 6.1% as Informative. The trend shifted towards Pro and Informative tweets globally as vaccination programs progressed, indicating a change in the public's perception of COVID-19 vaccines and vaccination. Furthermore, we explored the discourse based on account attributes, i.e., followers counts and tweet counts. Results show a significant pattern of discourse differences. Our findings highlight the potential of harnessing a large-scale geotagged Twitter dataset to understand global public health communication and to inform targeted interventions aimed at addressing vaccine hesitancy.