This paper interfaces combinatorial integral approximation strategies with the inherent robustness properties of conventional model predictive control with stabilizing terminal conditions. We deduce practical stability results for finite-control set and mixed-integer model predictive control and investigate the evolution of the closed-loop system in the presence of control rounding to draw conclusions about deviation from optimality. We propose an alternative Lyapunov function candidate and elaborate in detail the importance of the rounding history on the closed-loop performance. Sum-up rounding is one instance of an integer control reconstruction approach in combinatorial integral approximation. We show how to embed it into our theoretical findings to evaluate the rounding history and tightly bound the integral approximation error. Numerical experiments serve to illustrate the importance of an advanced rounding strategy in the context of mixed-integer model predictive control.