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Within the Asakura-Oosawa model, we study structural relaxation in mixtures of colloids and polymers subject to Brownian motion in the overdamped limit. We obtain the time evolution of the self and distinct parts of the van Hove distribution function G(r,t) by means of dynamical density functional theory (DDFT) using an accurate free-energy functional based on Rosenfeld's fundamental measure theory. In order to remove unphysical interactions within the self part, we extend the recently proposed quenched functional framework (Stopper et al 2015 J. Chem. Phys. 143 181105) toward mixtures. In addition, we obtain results for the long-time self diffusion coefficients of colloids and polymers from dynamic Monte Carlo simulations, which we incorporate into the DDFT. From the resulting DDFT equations we calculate G(r, t), which we find to agree very well with our simulations. In particular, we examine the influence of polymers which are slow relative to the colloids-a scenario for which both DDFT and simulation show a significant peak forming at r  =  0 in the colloid-colloid distribution function, akin to experimental findings involving gelation of colloidal suspensions. Moreover, we observe that, in the presence of slow polymers, the long-time self diffusivity of the colloids displays a maximum at an intermediate colloid packing fraction. This behavior is captured by a simple semi-empirical formula, which provides an excellent description of the data.