This paper interfaces combinatorial integral approximation strategies with the inherent robustness properties of conventional model predictive control with stabilizing terminal conditions to establish practical asymptotic stability results for finite-control set and mixed-integer model predictive control. We examine the impact of sequential control rounding on the closed-loop performance in terms of stability and optimality. Sum-up rounding takes its decision on the basis of the integrated control approximation error and represents one instance of a sequential integer control reconstruction approach in combinatorial integral approximation. We first apply outer convexification and relaxation and then show how to embed the class of sequential rounding algorithms into the step-by-step operation of model predictive control. Numerical experiments serve to illustrate the importance of an advanced rounding strategy in the context of mixed-integer model predictive control.