We found two three-component nonlocal conservation laws for the 3D rdDym equation, the universal hierarchy equation, the modified Veronese web equation, and the Veronese web equation. The aforementioned equations together with Pavlov’s equation are related via Bäcklund transformations. We study correspondences between the nonlocal conservation laws yielded by the Bäcklund transformations. In particular, we prove that the nonlocal conservation laws that depend on one pseudopotential are generated from a local conservation law of the Veronese web equation via appropriate superpositions of the Bäcklund transformations. Also, we prove nontriviality of the conservation laws found in the paper as well as the ones found in Makridin and Pavlov (2017) for the Khokhlov–Zabolotskaya equation and Pavlov’s equation.