We analyze the dynamics of a tracer particle embedded in a bath of hard spheres confined in a channel of varying section. By means of Brownian dynamics simulations, we apply a constant force on the tracer particle and discuss the dependence of its mobility on the relative magnitude of the external force with respect to the entropic force induced by the confinement. A simple theoretical one-dimensional model is also derived, where the contribution from particle-particle and particle-wall interactions is taken from simulations with no external force. Our results show that the mobility of the tracer is strongly affected by the confinement. The tracer velocity in the force direction has a maximum close to the neck of the channel, in agreement with the theory for small forces. Upon increasing the external force, the tracer is effectively confined to the central part of the channel and the velocity modulation decreases, which cannot be reproduced by the theory. This deviation marks the regime of validity of linear response. Surprisingly, when the channel section is not constant, the effective friction coefficient is reduced as compared to the case of a plane channel. The transversal velocity, which cannot be studied with our model, follows qualitatively the derivative of the channel section, in agreement with previous theoretical calculations for the tracer diffusivity in equilibrium. Published by AIP Publishing.