The EU's Green Deal proposal and Farm to Fork strategy explicitly call for both demand and supply measures to reduce food system emissions. While research clearly illustrates the importance of dietary transitions, impacts of potential supply-side measures are not well understood in relation to competitiveness concerns and leakage effects. This study assesses trade and GHG emission impacts of two supply-side mitigation strategies in the EU (plus UK and Switzerland), against a 2050 baseline featuring healthy/sustainable diets adopted by European consumers. To capture potential leakage effects arising from changing external trade flows, two supply-side strategies (intensification and extensification) are assessed against three trade policy regimes, resulting in six scenarios formulated with detailed inputs from the EUCalc model and simulated with a purported-designed CGE model. Our results show that intensification, while improving the EU+2's external trade balance, does not reduce its emissions, compared to the baseline. In contrast, extensification leads to a substantial emission abatement that augments reductions from the assumed dietary transition embodied in the baseline, resulting in a combined 31.1% of agricultural emission reduction in EU+2 during 2014-2050. However, this is at the expense of worsening agrifood trade balance amounting to US$25 billion, and significant carbon leakages at 48%, implying that half of the EU+2's emission reduction are cancelled out by rising emissions elsewhere. Furthermore, implementing the EU+2's prospective regional trade agreements leads to increased EU emissions; however, a border carbon adjustment by the EU+2 can improve its trade balance and partially shifting mitigation burdens to other countries, but ultimately only marginally reduce global emissions (and carbon leakage). Finally, different trade and emission effects are identified between the crop and livestock sectors, pointing to the desirability of a mixed agriculture system with intensified livestock sector and extensified crop agriculture in EU+2 that balances emission reduction goals and competitiveness concerns.