Quantum many-body systems with exact local gauge symmetries exhibit rich out-of-equilibrium physics such as constrained dynamics and disorder-free localization. In a joint submission, we present evidence of staircase prethermalization in a $Z_2$ lattice gauge theory subjected to a small breaking of gauge invariance. Here, we consolidate this finding and the associated emergent nonperturbative timescales analytically and numerically. By means of a Magnus expansion, we demonstrate how exact resonances between different gauge-invariant supersectors are the main reason behind the emergence of staircase prethermalization. Furthermore, we showcase the robustness of our conclusions against various initial conditions including different system sizes, matter fillings, and gauge-invariance sectors, in addition to various boundary conditions, such as different maximal on-site matter occupations. We also elaborate on how our conclusions are unique to local-symmetry models and why they break down in the case of global-symmetry breaking. We moreover extend our results to $U(1)$ lattice gauge theories, illustrating the generality of our findings. Our work offers an analytic footing into the constrained dynamics of lattice gauge theories and provides proof of a certain intrinsic robustness of gauge-theory dynamics to errors in experimental settings.