The EU’s ambition to establish economy-wide climate neutrality by 2050 requires challenging transformations in many economic activities. This paper aims to investigate the nexus of structural changes and greenhouse gas emissions (GHGEs) in an important sector of the livestock system, namely pig farming, during the period of 2010–2020 and to discuss the main directions of GHGE reduction. The academic novelty of this contribution is characterised by a combination of the shift-share and cluster analysis that allows for the investigation of the evolution phenomenon, applying the sustainability prism in order to understand the nexus between pig farming and the livestock system, as well as combining the national and EU levels. Results suggest that the steep decline in the number of holdings and a moderate reduction in livestock units (LSUs) on farms do not bring tangible GHGE reduction results. The cluster analysis confirms that pig farming systems in pre-2004 member states, except for Finland and Greece, demonstrated positive developments or a lower decline in holdings with pigs and live swine LSUs compared to other countries, while in the dominant share of post-2003 member states, the GHGE reduction rate was higher. This research identifies a reduction in the pig population, improvement in feed production and the development of related supply chains, and changes in manure management and utilisation as the main directions of GHGE reduction; however, the identified clusters are related with different potentials of GHGE reduction when applying the aforementioned measures. Recommendations include the development and support of actions that focus on GHGE reduction from swine manure and contribute to the establishment of a circular economy in the EU.