Within Density Functional Theory (DFT), we address the capture of a Cs atom by a superfluid helium nanodroplet using models of different complexity. In the simplest model, the Cs-droplet potential is obtained in two extreme approximations, namely the sudden approximation in which one assumes that the density of the droplet is not relaxed as Cs approaches it, and the adiabatic approximation in which one assumes that it does. Next, a more complex approach in which the collision is described within a time-dependent DFT approach is employed. Depending on the energy and impact parameter of the impinging Cs atom, a rich variety of dynamical phenomena appears that is discussed in some detail.