We develop a day-to-day route choice control method that is based on model predictive control (MPC). To influence the route choice of drivers we propose to use traffic control measures like variable speed limits or outflow control. In previous papers we have developed MPC for route choice control in the case of a constant demand. In this paper we consider the case of a time-varying demand. The resulting MPC optimization problem is in general nonlinear and nonconvex. However, in the case of outflow control and for a linear or a piecewise affine cost function it is possible to approximate the problem and to recast it as a mixed integer linear programming (MILP) problem, for which efficient branch-and-bound solvers are available. The solution of the MILP problem can then be used as a good initial starting point for a nonlinear optimization method for the original MPC optimization problem. We also illustrate the proposed approach for a simple simulation example involving outflow control.