<p>We develop a framework for studying consistent interactions of local gauge theories, which is based on the presymplectic BV-AKSZ formulation. The advantage of the proposed approach is that it operates in terms of finite-dimensional spaces and avoids working with quotient spaces such as local functionals or functionals modulo on-shell trivial ones. The structure that is being deformed is that of a presymplectic gauge PDE, which consists of a graded presymplectic structure and a compatible odd vector field. These are known to encode the Batalin–Vilkovisky (BV) formulation of a local gauge theory in terms of the finite dimensional supergeometrical object. Although in its present version the method is limited to interactions that do not deform the presymplectic structure and relies on some natural assumptions, it gives a remarkably simple way to analyse consistent interactions. The approach can be considered as the BV-AKSZ extension of the frame-like approach to consistent interactions. We also describe the underlying homological deformation theory, which turns out to be slightly unusual compared to the standard deformations of differential graded Lie algebras. As an illustration, the Chern-Simons and YM theories are rederived starting from their linearized versions.</p>