We study the impact of classical short-range nonlinear interactions on transport in lattices with no dispersion. The single-particle band structure of these lattices contains flat bands only, and cages noninteracting particles into compact localized eigenstates. We demonstrate that there always exist local unitary transformations that detangle such lattices into decoupled sites in dimension 1. Starting from a detangled representation and inverting the unitary transformations, we arrive at the all-bands-flat generator for single-particle Hamiltonians in one dimension, which is also straightforwardly generalized to higher dimensions. The entangling unitary transformations are parametrized by sets of angles. For a given member of the set of all-bands-flat, additional short-range nonlinear interactions destroy caging in general, and induce transport. However, fine-tuned subsets of the unitary transformations allow caging to be completely restored. We derive the necessary and sufficient fine-tuning conditions for nonlinear caging, and we provide computational evidence of our conclusions for one-dimensional systems.