From 1997 to 2002 he was director of Migrations Department of the Archdiocese of Buenos Aires, taking on in 1999 additional responsibilities for evangelization on behalf of the Pontifical Missions of Argentina.

[2] Beginning on 1 January 2017 he worked as one of the undersecretaries of the Migrants and Refugees Section of the Dicastery for Promoting Integral Human Development alongside Michael Czerny.

[5] On 23 April 2022, Pope Francis named him the sole undersecretary of his section and enlarged his responsibilities to include special projects.

[6] In 2019 he described his section's message:[1] So many people who arrive from outside, from cultures different from our own, from expressions of Christian or Catholic life different from ours, are a source of enrichment, and not of impoverishment, for us.

[8] On 31 October 2024, Pope Francis appointed Baggio as titular bishop of Urusi and gave him the personal title of archbishop.