We study the dynamics of soft colloids interacting via a Gaussian pair potential in an external moving potential which is periodic in the spatial coordinate of the direction of motion. Both dynamical density functional theory and Brownian dynamics computer simulations are used to predict the steady-state density profiles. Two different situations are investigated: the first corresponds to a light wave that travels with a constant velocity v through the quiescent solvent containing the colloidal suspension. The second setup consists of two parallel repulsive walls with a periodic topographical substructure. One of the walls is at rest relative to the solvent while the other is in motion, inducing a shearing of the suspension. In the first case, we find that the amplitude of the steady-state density behaves nonmonotonically with the traveling speed v of the wave if the shape of the wave contains an edge: for increasing v , it first grows and then decreases. In the second setup we show that a strongly confined suspension induces a shear resistance which is a nonmonotonic function of the wall velocity. These effects are verifiable in real-space experiments on colloidal suspensions exposed to external laser-optical fields. In both situations, the dynamical density functional theory is in good agreement with the Brownian dynamics simulation data.