Anti-vaccination views pervade online social media, fueling distrust in scientific expertise and increasing vaccine-hesitant individuals. While previous studies focused on specific countries, the COVID-19 pandemic brought the vaccination discourse worldwide, underpinning the need to tackle low-credible information flows on a global scale to design effective countermeasures. Here, we leverage 316 million vaccine-related Twitter messages in 18 languages, from October 2019 to March 2021, to quantify misinformation flows between users exposed to anti-vaccination (no-vax) content. We find that, during the pandemic, no-vax communities became more central in the country-specific debates and their cross-border connections strengthened, revealing a global Twitter anti-vaccination network. U.S. users are central in this network, while Russian users also become net exporters of misinformation during vaccination roll-out. Interestingly, we find that Twitter's content moderation efforts, and in particular the suspension of users following the January 6th U.S. Capitol attack, had a worldwide impact in reducing misinformation spread about vaccines. These findings may help public health institutions and social media platforms to mitigate the spread of health-related, low-credible information by revealing vulnerable online communities.