[3] Barragán went on to study clarinet at the Conservatorio Superior de Música Manuel Castillo in Sevilla with Antonio Salguero and at the Barenboim-Said Foundation's Seville Academy with Matthias Glander.

[6] Barragán has taught as professor at the Barenboim-Said Academy in Seville from 2020; he has held master classes including at the Reina Sofía School of Music in Madrid.

[9] Barragán's chamber music partners have included violinist Liza Ferschtman, cellist Kian Soltani, pianists Elena Bashkirova and Beatrice Rana, flutist Emmanuel Pahud and the Modigliani Quartet.

[5] After playing the Brahms Clarinet Quintet in a 2023 concert with the Schumann Quartet at London's Wigmore Hall, a reviewer noted that they began playing "poised and searching", with the clarinet as "first among equals", then expressed "sweet melancholy and resurgent passion" in the slow movement and "all the complexity of a human being's warring passions" in the final movement, in a "fully involving performance".

[12][13] He recorded chamber music of the 20th century in 2024 with violinist Noa Wildschut and pianist Frank Dupree, including Paul Schoenfield's trio Refractions and Claude Vivier's Pièce for violin and clarinet.