We study the higher-derivative equations for gauge potentials of arbitrary mixed-symmetry type obtained by setting to zero the divergences of the corresponding curvature tensors. We show that they propagate the same reducible multiplets as the Maxwell-like second-order equations for gauge fields subject to constrained gauge transformations. As an additional output of our analysis, we provide a streamlined presentation of the Ricci-like case, where the traces of the same curvature tensors are set to zero, and we present a simple algebraic evaluation of the particle content associated with the Labastida and with the Maxwell-like second-order equations.