Abstract <p>In this paper, for the first time, a numerical continuation algorithm for dynamical integrity assessment—also known as basin stability—is developed. The method exploits information about the dynamical integrity of previous parameter sets—computed through a <Emphasis FontCategory="NonProportional">DynIn</Emphasis> based algorithm—for estimating the local integrity measure of a neighboring parameter set, whose value is then corrected. Overall, the algorithm resembles a classical predictor-corrector scheme used for numerical continuation of steady-state solutions, but it is applied to the local integrity measure. The method is tested on three diverse dynamical systems, namely, a non-autonomous Duffing oscillator, a van der Pol-Duffing oscillator with an attached vibration absorber (autonomous system), and an industrial model of a self-synchronizing vibrating screen. Despite the challenging scenarios encountered, encompassing fractal basins of attraction, several coexisting stable solutions and moderately large dimensional systems, the method was able to rapidly and accurately generate robustness maps for all of them, illustrating the systems’ dynamical integrity for large ranges of the parameter sets. The algorithm also unveiled branches of isolated periodic solutions, which would be unattainable by traditional continuation methods. Additionally, we extended the capabilities of the <Emphasis FontCategory="NonProportional">DynIn</Emphasis> toolbox, enabling it to handle chaotic solutions and to efficiently compute basins of attraction sections.</p> Graphic Abstract <p></p>